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joaogui1 (he/him) (Mar 01 2021 at 10:14):I'm starting to learn category learning and I have a couple of questions on books and beyond. Right now I'm in two reading groups, one on "Category Theory for Programmers" and another on "Seven Sketeches in Compositionality".
Matteo Capucci (he/him) (Mar 01 2021 at 10:20):Uh, there's certainly a lot of overlap among these books. I didn't read any of them from cover to cover though, so I don't know exactly how much. What's sure is that each of them covers CT from a different point of view, and this means that the set of things you'll learn from each of them is different, and even in the intersection the way you're exposed to thing is gonna be different.
For example, monads in Milewski's book are gonna be explained from a CS point of view (effects or inductive data types), which is quite different from monads in algebra (free constructions) and monads in topology (completions, I guess?).
This is a general theme in CT: the core concepts are not numerous, but they are incredibly versatile. So learning is not idempotent anymore.
Fawzi Hreiki (Mar 01 2021 at 10:57):You can read 'Topology: A Categorical Approach' without reading any Munkres. They're both introductions to point-set topology but with slightly different approaches. 'Algebra: Chapter 0' is a fantastic graduate algebra book so that's a good place to start.
Fawzi Hreiki (Mar 01 2021 at 10:58):If you want a slightly more advanced category theory book, Emily Riehl's 'Category Theory in Context' is great and also very cheaply available from Dover.
Jules Hedges (Mar 01 2021 at 11:06):To answer 4, by the time you've read all those textbooks you've already read twice as many category theory textbooks as I ever have
Fawzi Hreiki (Mar 01 2021 at 11:06):One point: don't worry too much about getting through books cover to cover. Its usually not an efficient way to learn
Jules Hedges (Mar 01 2021 at 11:10):This is a very personal thing, but personally I find that early on picking some research question (even if it turns out to be a bad research question in retrospect) is a much more effective way for me to learn than just reading stuff
John Baez (Mar 01 2021 at 16:45):Hi! Conceptual Mathematics is a very distinctive book: it explains Lawvere's philosophy of category theory in a way aimed at beginners. (Lawvere, you may have heard, had a lot of important ideas in category theory.) So, no other book will replace that book, but also it's not what I'd call a "reference" - it's more like an adventure of its own.
John Baez (Mar 01 2021 at 16:50):Fawzi Hreiki said:
If you want a slightly more advanced category theory book, Emily Riehl's 'Category Theory in Context' is great and also very cheaply available from Dover.
I'd say that's more than "slightly" more advanced than the other books listed. It's aimed at people who know about lots of subjects in math, and it draws its examples from those. But it's a great book to read after those listed if you're into math.
Mike Shulman (Mar 01 2021 at 17:35):Personally, I can't imagine learning category theory from Conceptual mathematics. But once you already know what it's talking about, it's an amusing read. YMMV of course.
John Baez (Mar 01 2021 at 17:48):I think beginners could learn something from Conceptual Mathematics, but it would be something more like Lawvere's attitude toward math. (I don't, in fact, know any beginner who succeeded in getting through this book.)
John Baez (Mar 01 2021 at 17:48):Lawvere would be pretty pissed off if he heard you call it "an amusing read".
Fawzi Hreiki (Mar 01 2021 at 18:13):I guess a step down in difficulty from Emily's book would be Tom Leinster's but that's still aimed at math students rather than CS or engineering people.
Fawzi Hreiki (Mar 01 2021 at 18:17):Content-wise, there isn't that much to learn in a first category theory course, but it takes a while to actually internalise the concepts so it helps to relearn the same thing multiple times from different perspectives.
Fawzi Hreiki (Mar 01 2021 at 18:26):Especially since so many great books and lecture notes are available online for free.
Kale Evans (Mar 14 2021 at 06:44):Newbie question..do any two categories have at least one functor between them? Are there pairs of categories that have no functors?
Kale Evans (Mar 14 2021 at 06:45):I googled this, and thought I'd easily find some theorem that stated one of these concepts, but no dice so far. I'm still in the beginning stages of learning, and sure I'd find this result eventually and naturally, but I wanted to check in on this now
John Baez (Mar 14 2021 at 07:06):If is a category with at least one object and has no objects, then there is no functor .
But if is a category with at least one object and is any category at all, there's at least one functor . We can get one by choosing an object , and mapping every object of to , and every morphism in to the morphism . You can check that this gives a functor.
John Baez (Mar 14 2021 at 07:07):Also, if both and have no objects, then there is a unique functor from .
John Baez (Mar 14 2021 at 07:11):So, for any two categories and , there is always a functor or a functor (or both).
John Baez (Mar 14 2021 at 07:12):All this is very like functions between sets: there's no function from a nonempty set to an empty set, but there's always a function from a nonempty set to any other nonempty set, and there's a unique function between empty sets.
Kale Evans (Mar 14 2021 at 22:01):So interesting and so well explained! Thank you so much @John Baez .
So what are the implications of this?
If I have two categories, from two completely disparate scientific disciplines For example, and a functor between them, that of course by definition, preserves the original relations within the initial category, have you in essence discovered a way of describing one sceintific discipline in terms of another?
Or does this kind of benefit depend on the functor itself? For example, I can't imagine a trivial functor such as one that maps every object in the original category to a single object in the other, and every morphism to the identity, as one that introduces a new way of discussing relations and concepts between the two categories.
Or am I being completely idealistic, and a functor doesn't necessarily give you a way of relating scientific disciplines, or in general, discussing one category in terms of another?
John Baez (Mar 14 2021 at 22:04):My example of how to get a functor from any category to any nonempty category is pretty boring. In terms of relating scientific disciplines, supposed you used this to relate geology to ornithology. Then it would be saying "every rock is kind of like a penguin, and every way to turn one rock into another is kind of like doing nothing to a penguin".
Kale Evans (Apr 05 2021 at 03:11):D8CAE705-ADEB-45D3-A9F6-0199AE241FC4.jpg
Kale Evans (Apr 05 2021 at 03:11):I'm reading about adjunctions for the first time and having trouble seeing what functors fulfill the naturality requirements for the isomorphism of hom-sets. For Categories C and D, it would appear it would be the functors C-> Set and D-> Set are natural. Though Dr. Fong and Dr. Spivak's book seems to suggest it's C_op -> Set and D -> Set. But then again, the commutative diagram in the book seems to suggest otherwise.
Kale Evans (Apr 05 2021 at 03:13):So for Categories C and D, and functors L: C ->D and R: D->C, which functors commute when L and R are left and right adjoints respectively?
Kale Evans (Apr 05 2021 at 03:59):A follow-up question. Can functors that have a natural transformation between them be considered equivalent? I hadn't thought about it in this way, but looking at curry adjunction example in Dr Fong and Spivak's book seems to suggest this.
Ralph Sarkis (Apr 05 2021 at 07:10):Kale Evans said:
So for Categories C and D, and functors L: C ->D and R: D->C, which functors commute when L and R are left and right adjoints respectively?
You should have seen at some point the covariant and contravariant hom functors and . These can be combined into a hom bifunctor acting on objects by and on morphisms by . You can in turn combine this bifunctor with or to obtain the relevant functors. Here is the definition in my notes (work in progress):
image.png
Ralph Sarkis (Apr 05 2021 at 07:13):Apologies for the different notation, you can change and with and respectively.
Ralph Sarkis (Apr 05 2021 at 07:13):Kale Evans said:
A follow-up question. Can functors that have a natural transformation between them be considered equivalent? I hadn't thought about it in this way, but looking at curry adjunction example in Dr Fong and Spivak's book seems to suggest this.
This is true when you have a natural isomorphism. Indeed, these are the isomorphisms of functor categories so it would be evil not to consider two naturally isomorphic functors equivalent.
John Baez (Apr 05 2021 at 15:20):More bluntly, the answer to this question:
Can functors that have a natural transformation between them be considered equivalent?
is no, unless the natural transformation is an natural isomorphism. This question is like asking "should sets with a function between them be considered equivalent?" Indeed, the latter question is a special case of the former, since a set is the same as a functor , and a function between sets is the same as a natural transformation between such functors. (Here is the category with just one object and its identity morphism.)
John Baez (Apr 05 2021 at 15:21):The point is: things should not be considered equivalent unless there is an invertible map between them; an invertible natural transformation is the same as a natural isomorphism.
Mike Stay (Apr 05 2021 at 20:15):things should not be considered equivalent unless there is an invertible map between them
@Kale Evans John's using n-categorical language when he says "equivalent" here. In the category theory literature, two objects are "isomorphic" when there's an invertible map between them. The word "equivalent" applies in a more general scenario. For example, objects in a 2-category are equivalent if there's a pair of morphisms but isn't equal to the identity on , only isomorphic to it. Similarly, is isomorphic to the identity on . So in this case, "equivalent" means "isomorphic up to 2-isomorphism". When talking about n-categories, they'll say that two objects in an n-category are "equivalent" if they're "isomorphic up to 2-isomorphism up to ... up to n-isomorphism". So while it's technically true that two sets can be equivalent, or even two elements of a set, people usually use the terms "isomorphic" and "equal" for those cases.
Kale Evans (Apr 06 2021 at 06:02):Thank you guys so much for your replies! @Ralph Sarkis I have to think on your answer for a bit. I haven't been introduced to covariant and contravariant him functors yet. So I'm reading up on your reply:smile:. Though it actually does help clarify my suspicions that my thinking was a bit off with regards to this. Maybe a follow-up question will come, but for now, I'll ponder.
Kale Evans (Apr 06 2021 at 06:05):@Mike Stay thank you and @John Baez I always really appreciate the time you always put in to explicitly answer questions from the community, of course me included. Very big thank you :pray:🏻
@Mike Stay your clarification between isomorphic and equal really really helps. I was having trouble seeing how the conclusion of equivalency made sense with this result. I was applying toy examples and finding that two isomorphic objects were not actually equivalent.
Kale Evans (Apr 06 2021 at 06:06):For example, a category of real numbers with morphisms being maps between them. 1 and 2 have invertible maps between them, but are not equal, and these kind of examples were causing me confusion.
Kale Evans (Apr 06 2021 at 06:11):But according to your reply, they can be seen as isomorphic but not equivalent. Is that right?
I guess isomorphism is most useful for functors and functions in particular, given the result shows that isomorphic functions can net the same result when applied in a particular fashion or composed with another function I suppose.
This is an interesting result that I'm interested in seeing how it is and can be applied in real-world applications. So I wanted to check that my assumptions were sound.
Morgan Rogers (he/him) (Apr 06 2021 at 07:57):Kale Evans said:
For example, a category of real numbers with morphisms being maps between them. 1 and 2 have invertible maps between them, but are not equal, and these kind of examples were causing me confusion.
Be careful: what exactly is the category you're describing here? Is a category whose objects are real numbers, and with one morphism between any pair of objects? That's what it sounds like from the conclusion, but it doesn't seem like the category you were describing, since these aren't really "maps between real numbers". Is it the category with a single object, with morphisms being endomorphisms of the reals?
Morgan Rogers (he/him) (Apr 06 2021 at 08:00):Kale Evans said:
But according to your reply, they can be seen as isomorphic but not equivalent. Is that right?
No, isomorphism is a special case of equivalence, so any isomorphism is a kind of equivalence.
I guess isomorphism is most useful for functors and functions in particular, given the result shows that isomorphic functions can net the same result when applied in a particular fashion or composed with another function I suppose.
Isomorphism is the kind of equivalence that appears between objects within 1-categories (aka ordinary categories). If two objects are isomorphic, they will have the same relationship to all other objects in the category.
John Baez (Apr 06 2021 at 15:10):Kale Evans said:
For example, a category of real numbers with morphisms being maps between them. 1 and 2 have invertible maps between them, but are not equal, and these kind of examples were causing me confusion.
There's no such thing as a map between real numbers. There's a function between the set and the set , and it has an inverse, so these sets are isomorphic. But there's no such thing as a function between 1 and 2.
Of course you can invent your own concept of maps between real numbers, and you can say that there's a unique map from any real number to any other real number - but you've just shown that this is not a very useful concept, because it makes all real numbers isomorphic and doesn't do anything very interesting.
Kale Evans said:
But according to your reply, they can be seen as isomorphic but not equivalent. Is that right?
Mike was trying to sketch out a precise concept of "equivalence" that's used in category theory and higher category theory. That concept has a precise definition. Mathematicians also use "equivalent" in a flexible, loose way where you say two things are equivalent if they're the same in any way you're interested in. I think you are using equivalence in the second way. In the second way, you have a lot of freedom in deciding what's equivalent to what: it depends on what you're doing.
Mike Stay (Apr 06 2021 at 15:47):John Baez said:
Of course you can invent your own concept of maps between real numbers
One popular way to do that with real numbers is to say there's a morphism from to if but then the only time numbers are isomorphic is when they're equal.
There's also a category with one object = the set of real numbers and lots of morphisms from that object to itself, the functions from to . Infinitely many of these take 1 to 2 and vice versa, but 1 and 2 are no longer objects of the category, they're elements of the one object of the category.
You could promote them from mere elements to objects by doing the category of elements construction to Objects are real numbers and morphisms from to are functions such that but as @John Baez said, that makes all real numbers isomorphic and isn't very interesting.
Kale Evans (Apr 07 2021 at 04:54):No, isomorphism is a special case of equivalence, so any isomorphism is a kind of equivalence.
I see. My misunderstanding. I suppose I was trying to relate isomorphisms in category theory to equivalences in other fields, and that is not the right approach for sure.
I suppose it is my lack of experience that currently prevents me from seeing how isomorphic objects in a category are seen as equivalent as they relate to other objects in a category. I haven't seen an example that really illuminates that concept yet. So I was erroneously attempting to tease out special characteristics by creating a category that would model objects in concrete math that I'm more familiar with. Namely, the reals. But this is not the right approach. I think I'll have to continue to work thru examples to gain intuition of how isomorphic objects are usuful with respect to other objects in their Categories
Kale Evans (Apr 07 2021 at 05:01):There's no such thing as a map between real numbers. There's a function between the set and the set , and it has an inverse, so these sets are isomorphic. But there's no such thing as a function between 1 and 2.
This was a very sloppy example of mine.:persevere: Thank you for the clarification.
Mike was trying to sketch out a precise concept of "equivalence" that's used in category theory and higher category theory. That concept has a precise definition. Mathematicians also use "equivalent" in a flexible, loose way where you say two things are equivalent if they're the same in any way you're interested in. I think you are using equivalence in the second way. In the second way, you have a lot of freedom in deciding what's equivalent to what: it depends on what you're doing.
That helps me understand the usages of the term @John Baez , thank you. I will continue my study to see how isomorphic objects can be seen as equivalent in the typical categorical sense of the term. I feel I'm starting to find that this is something you get a feel for with more practice. Most texts introduce the concept of isomorphism, but without an adjoining example of how isomorphims can be used, and how they should be appreciated. So I'm on the road to appreciating this property, and these comments have helped put things in perspective for me:smile:
John Baez (Apr 07 2021 at 05:35):Most mathematicians become familiar with isomorphisms of vector spaces, isomorphisms of groups, isomorphisms of rings, isomorphisms of topological spaces, and isomorphisms of objects in various other famous categories before meeting the concept of isomorphism in its full abstract generality. So, when they read a book on category theory, they already know how isomorphisms work in a bunch of examples: they know instinctively that isomorphic objects are "the same in every important way".
John Baez (Apr 07 2021 at 05:36):This, at least, is the hope of most mathematicians who write on category theory!
Morgan Rogers (he/him) (Apr 08 2021 at 09:56):Kale Evans said:
No, isomorphism is a special case of equivalence, so any isomorphism is a kind of equivalence.
I see. My misunderstanding. I suppose I was trying to relate isomorphisms in category theory to equivalences in other fields, and that is not the right approach for sure.
That sounds like a very sensible approach!
I suppose it is my lack of experience that currently prevents me from seeing how isomorphic objects in a category are seen as equivalent as they relate to other objects in a category.
Suppose I have an isomorphism with inverse . Whenever I have a morphism , which can be thought of as a mapping, or more generally a relationship from to , then by composing with , I get a relationship from to for free. Similarly, by composing with , any relationship from to produces a relationship from to . The fact that and compose to give identities in both directions mean that these operations of composition are exactly inverses to one another: there's a bijective correspondence between morphisms and . This is also true on the other side: there's a bijection between morphisms and . Since morphisms into and out of are the only way that other objects in the category can "see" , the consequence of this is that and 'look identical' from the perspective of any object in the category.
Morgan Rogers (he/him) (Apr 08 2021 at 09:57):If you tell me your favourite category, I can make this idea of "seeing via morphisms" more precise. :grinning_face_with_smiling_eyes:
Kale Evans (Apr 09 2021 at 02:06):The fact that and compose to give identities in both directions mean that these operations of composition are exactly inverses to one another: there's a bijective correspondence between morphisms and .
Can you help me here @Morgan Rogers (he/him) ? I'm failing to see how morphisms and are inverses of each other. I thought for morphisms to be inverses of each other they had to be composable, and it appears and aren't
John Baez (Apr 09 2021 at 02:08):He didn't say that morphisms and were inverses of each other.
Kale Evans (Apr 09 2021 at 02:10):Oh, I thought that's what the following meant.
these operations of composition are exactly inverses to each other
John Baez (Apr 09 2021 at 02:10):He said you can turn any morphism into a morphism , and also you can turn any morphism back into a morphism . (He said it with more words and fewer symbols.)
John Baez (Apr 09 2021 at 02:10):The operation sending to and the operation sending to are exactly inverses to each other.
John Baez (Apr 09 2021 at 02:12):For example: start with , apply the first operation and get , and then apply the second operation and get . But so you're back where you started!
John Baez (Apr 09 2021 at 02:13):Thus, an isomorphism between and gives a bijection between morphisms from to and morphisms from to .
John Baez (Apr 09 2021 at 02:14):This is one of many senses in which isomorphic objects "act the same".
Kale Evans (Apr 09 2021 at 02:47):Ahh I see! The operation of sending the starting morphism to the composition are inverses. That makes total sense. This clears it up for me. Thank you so much @Morgan Rogers (he/him) and @John Baez for the speedy and thorough replies. :big_smile:
Kale Evans (Jul 22 2021 at 05:41):6B3CF71F-6073-4844-A6BE-FCEB1157D482.jpg
Kale Evans (Jul 22 2021 at 05:43):I had a question about compact closed categories and wire daigrams.
What is an intuitive understanding of what's happening in wire diagrams, representing a compact closed category? I understand that these categories have unit and counit morphisms that, when composed, are the identity morphism.
These morphisms are analogous to the cap and cup icons in a wiring diagram. So how can one interpret the diagrams with respect to the analogy?
Take the attached image. There's a cap and cup between the motor and the summation morphisms that represents weight. What exactly does that mean? That the morphisms between them amount to the identity morphism on weight?
Kale Evans (Jul 22 2021 at 05:43):.
Jérémie Koenig (Jul 23 2021 at 00:52):@Kale Evans my intuition is that between and information flows in opposite directions (cf. the and annotations in your diagram). Then the caps and cups allow you to reverse the flow of information but without any other transformation, hence if you reverse it twice you get back the identity morphism.
As you say, "the morphisms between [motor/battery and Σ] amount to the identity morphisms" in the following way: modulo some braiding, you could move the summation box to the right so that all of its input are fed directly from the left (via identity morphisms), and then if you use caps and cups to feed back the output of "Σ" to the "Chassis" box, you would get an equivalent diagram overall. Or you could flip Σ around 180 degrees (take its dual Σ*) and put it on top of the diagram, etc.
Kale Evans (Jul 23 2021 at 02:34):@Jérémie Koenig thank you! As weird as it sounds, I hadn't thought of caps and cups reversing the flow of information. I still considered them directional for some reason. That actually helps my intuition alot!
Kale Evans (Aug 02 2021 at 17:07):Am I right in thinking the defining characteristic difference between a symmetric monoidal Category and a commutative monoidal category is the latter's associator, unitor, and symmetric/commutative properties are natural identity morphisms as opposed to just natural isomorphisms?
I understand commutative monoidal categories are a special case of symmetric monoidal ones; I'm trying to get a good understanding of the additional requirements for the commutative ones
Nathanael Arkor (Aug 02 2021 at 17:19):Yes, that is correct: see the nLab page on commutative monoidal categories, for instance.
John Baez (Aug 03 2021 at 02:20):There are many equivalent ways to define commutative monoidal categories. You gave one. Another one is that a commutative monoidal category is a commutative monoid object in Cat: that is, a category with a "multiplication"
and "unit"
obeying the usual laws of a commutative monoid, written out in diagrams.
Another equivalent way to define commutative monoidal categories is that they are category objects in CommMon. The nLab article spells this out.
John Baez (Aug 03 2021 at 02:23):My paper with Jade Master spells out a few equivalent definitions.
Patrick Nicodemus (Aug 04 2021 at 20:55):John Baez said:
There are many equivalent ways to define commutative monoidal categories. You gave one. Another one is that a commutative monoidal category is a commutative monoid object in Cat: that is, a category with a "multiplication"
and "unit"
obeying the usual laws of a commutative monoid, written out in diagrams.
Another equivalent way to define commutative monoidal categories is that they are category objects in CommMon. The nLab article spells this out.
related question: when we say that a monoidal category is a monoid in Cat, what is the precise definition of monoid that we are using here?, and what is the technical term that we use to describe it in full? I guess we are viewing Cat as a 2-category here but one must throw the prefixes 'pseudo' or 'lax' in there somewhere and I am not sure in general what the right terminology is to describe the coherence conditions.
Nathanael Arkor (Aug 04 2021 at 21:34):Since everything is strict, we consider Cat as a (large) cartesian monoidal 1-category, and then a commutative monoid object is just as is defined in any cartesian monoidal category.
Kale Evans (Aug 05 2021 at 01:46):A follow up question since CommMon was brought up. I recently read that the monoid homomorphisms in CommMom form a commutative monoid. Though I'm having trouble finding this derivation anywhere. Does anyone know where I can find an explanation, or perhaps can provide a brief one?
John Baez (Aug 05 2021 at 04:51):Here's how it works. Suppose are monoid homomorphisms and is commutative. Then we can "multiply" and and get monoid homomorphism which I will call . Here's how:
Here's the proof that is really a monoid homomorphism:
Note that in the first equation we used the fact that is commutative. We did not need to be commutative.
It's pretty easy to check that this multiplication of homomorphisms from to makes the set of these homomorphisms into a commutative monoid. In other words:
and
where is the homomorphism defined by
I leave these verifications to you!
Kale Evans (Aug 05 2021 at 21:42):In the first equation, the result should read right?
John Baez (Aug 05 2021 at 21:43):Yes. I'll fix it.
Kale Evans (Aug 05 2021 at 21:45):Ok thank you for the explanation! Gonna work on verification
John Baez (Aug 05 2021 at 21:45):I hope you see this formula for is the usual way we define how to multiply two real-valued functions, like . (Or, for that matter, add them, if the monoid operation in is called addition.)
John Baez (Aug 05 2021 at 21:45):So, it's very common to make the set of functions from any set to a commutative monoid into a commutative monoid using this formula!
John Baez (Aug 05 2021 at 21:47):But the small extra twist is that if is a monoid, the set of monoid homomorphisms is closed under this kind of multiplication.
Kale Evans (Aug 06 2021 at 04:28):Yeah, the definition actually threw me off at first, bc I'm used to seeing that for functions, but not morphisms. My assumption as to why it works is because a monoid is a set, so the morphisms between them are functions, and the tensor product of functions is defined in this way is my assumption.
John Baez (Aug 06 2021 at 04:46):Kale Evans said:
Yeah, the result actually threw me off at first...
That's not a "result" - it's a definition.
John Baez (Aug 06 2021 at 04:47):A "result" is something that follows from something, while a definition is just something you're free to make.
John Baez (Aug 06 2021 at 05:08):My assumption as to why it works is because a monoid is a set, so the morphisms between them are functions,...
Right: when we want to be fussy we say a monoid has an underlying set, or it's a set with extra structure. Similarly a morphism between monoids has an underlying function, or it's a function with extra properties.
John Baez (Aug 06 2021 at 05:09):The thing I defined is not a "tensor product" of functions, it's a way to multiply two functions taking values in a commutative monoid and get another function valued in that commutative monoid.
John Baez (Aug 06 2021 at 05:09):"Tensor product" means something a bit different.
Kale Evans (Aug 06 2021 at 14:05):That makes sense. I was sloppy with my description. Updated my comment. With regard to "tensor product" , I feel I've seen it referred to as that before maybe, but I mean the monoidal product, which is the correct way to describe it right?
In string diagrams, which are often used to represent monoidal categories, I've heard the "product" or two objects or morphisms described as a "tensor". This is the same operation right?
John Baez (Aug 06 2021 at 14:34):We say a category has a "tensor product", in the most general sense, when there's a way to take two morphisms , in and combine them somehow to get a morphism
where is some object built from and , and similarly for . For details, read about monoidal categories.
So, this is different than what you were talking about, which is a way of taking two morphisms , and combining them to get a morphism .
John Baez (Aug 06 2021 at 15:00):Both these ideas are very important and have been studied a lot.
Kale Evans (Aug 06 2021 at 17:56):I see. Ok, Yeah, you're right. Very different concepts; I conflated the two, given I thought the only operation in a monoidal category was the monoidal product which apparently is also referred to as a tensor product, according to wiki anyway and some professors I've seen on video.
Is there a specific flavor of monoidal category that has this operation 'combining' or 'multiplying' two objects or morphisms defined, if it's not considered the monoidal product? Or is it just another way of defining two morpshims with same source target for example? Or, are they just the same operation given the category?
Also, just to confirm, the tensor you referred to would be right?
Lastly, I have one maybe interesting question, maybe not..
but assuming we have a monoidal category , with finite products, and two morphisms .
And we have a tensor, representing the monoidal product,
But we also have a way of combining or 'multiplying' to get ( not sure if these dual concepts can exist in the same monoidal category).
If the above is possible, then we would have morphisms and . And potentially a commutative square as in attached image? Screen-Shot-2021-08-06-at-1.56.03-PM.png
Keith Elliott Peterson (Aug 08 2021 at 02:52):In , the size of the internal hom, , is known to be , however, we category theorists tend to work up to iso.
But after reading on the 12fold way something bothers me about this: does the size of hom-sets not change when thinking up to iso?
John Baez (Aug 08 2021 at 02:59):Kale Evans said:
.... I thought the only operation in a monoidal category was the monoidal product which apparently is also referred to as a tensor product, according to wiki anyway and some professors I've seen on video.
Yeah, a monoidal category comes with a functor that's called the "tensor product" or "monoidal product". I think only hardcore category theorists say "monoidal product"; lots of people say "tensor product", but "tensor product" is often used by normal mathematicians (not category theorists) to mean the monoidal product in categories with a linear algebra flavor, like Vect or the category of modules of a commutative ring.
John Baez (Aug 08 2021 at 03:01):Is there a specific flavor of monoidal category that has this operation 'combining' or 'multiplying' two objects or morphisms defined, if it's not considered the monoidal product?
I don't understand that question. Maybe the problem is that I don't understand what "this operation" refers to. Which operation?
Kale Evans (Aug 08 2021 at 03:03):The 'combine' or 'multiply' operation for the two homomorphisms and resulting in . For this particular Category, is this simply the monoidal product?
John Baez (Aug 08 2021 at 03:04):Kale Evans said:
Also, just to confirm, the tensor you referred to would be right?
Yeah, there was a typo in my comment, which I've fixed now.
Lastly, I have one maybe interesting question, maybe not.
but assuming we have a monoidal category , with finite products, and two morphisms .
And we have a tensor, representing the monoidal product,
But we also have a way of combining or 'multiplying' to get ( not sure if these dual concepts can exist in the same monoidal category).
If the above is possible, then we would have morphisms and .
Why do you say we would have those? I don't see how you're getting them from the stuff you've posited. Nothing you posited gives a morphism from to , does it?
John Baez (Aug 08 2021 at 03:07):There certainly exist categories that are both:
and
John Baez (Aug 08 2021 at 03:09):An example is the category of commutative monoids: we can tensor commutative monoids, but also multiply commutative monoid homomorphisms.
John Baez (Aug 08 2021 at 03:11):If you don't know what "enriched over the category of monoids" means, you may (or may not) want to learn a bit about enriched categories: that would make this idea precise.
Kale Evans (Aug 08 2021 at 03:12):Why do you say we would have those? I don't see how you're getting them from the stuff you've posited. Nothing you posited gives a morphism from to , does it?
Oh, I meant assuming the was the actual product operation, then by definition
Kale Evans (Aug 08 2021 at 03:14):There certainly exist categories that are both:
- monoidal, so that given morphisms we can tensor them and get a morphism
and
- enriched over the category of monoids, so that given morphisms we can multiply them and get a morphism .
Oh very interesting. :thinking: Primordial ooze strikes again
Kale Evans (Aug 08 2021 at 04:00):I love it :big_smile:
John Baez (Aug 08 2021 at 07:06):Kale Evans said:
Why do you say we would have those? I don't see how you're getting them from the stuff you've posited. Nothing you posited gives a morphism from to , does it?
Oh, I meant assuming the was the actual product operation, then by definition.
I don't know what that means. In category theory the product of an object with itself, , comes with a morphism called the diagonal, . The coproduct comes with a morphism called the codiagonal, .
Reid Barton (Aug 08 2021 at 12:21):By the way, if you're working with categories enriched in commutative monoids, it's probably more helpful to write those monoids additively. Then you can distinguish the monoidal operation from the composition (and have the usual distributive laws like , etc.)
John Baez (Aug 08 2021 at 15:54):Right. It's my fault for starting to write the monoid structure as in this conversation. I was in a multiplicative mood.
(By the way, @Kale Evans: when Reid says "the monoidal operation" he is not talking about monoidal categories. I would say "the monoid operation" to avoid this potential confusion, though it's obvious from context what Reid means.)
Mike Shulman (Aug 08 2021 at 16:19):I don't know whether this will be helpful or more confusing, but there is a connection between monoidal structure and commutative-monoid enrichment. Namely, if the monoidal structure of a category is a biproduct, i.e. simultaneously a cartesian product and a cartesian coproduct, then it automatically becomes enriched over commutative monoids — and conversely, if a category is enriched over commutative monoids, then if it has products or coproducts they coincide and are biproducts.
In one direction, let's write for a biproduct, which therefore has both a diagonal and codiagonal . Then we can define a commutative-monoid operation on morphisms as the composite . Conversely, in a category enriched over commutative monoids, given a product we have the zero morphism (the commutative monoid identity) , where denotes the terminal object, and therefore a morphism ; similarly we have a morphism and we can show that these are the coprojections making also a coproduct.
John Baez (Aug 08 2021 at 17:35):That's cool. I'm not sure I ever knew this. I got excited 11 years ago when I noticed that every object in a category with biproducts becomes a bimonoid.
Kale Evans (Aug 08 2021 at 18:54):John Baez said:
Kale Evans said:
Why do you say we would have those? I don't see how you're getting them from the stuff you've posited. Nothing you posited gives a morphism from to , does it?
Oh, I meant assuming the was the actual product operation, then by definition.
I don't know what that means. In category theory the product of an object with itself, , comes with a morphism called the diagonal, . The coproduct comes with a morphism called the codiagonal, .
In addition to the diagonal, , don't we also get a morphism . Either of the or morphisms as shown in this diagram Screen-Shot-2021-08-08-at-2.52.16-PM.png ?
Kale Evans (Aug 08 2021 at 18:55):Or the morphisms as its referred to on nlab? Screen-Shot-2021-08-08-at-2.55.09-PM.png
John Baez (Aug 09 2021 at 04:18):You get those projections when the tensor product is the cartesian product. If you're assuming that, you should write rather than . Since you wrote I thought you were talking about a general monoidal category, not a cartesian one.
Morgan Rogers (he/him) (Aug 09 2021 at 08:03):Keith Peterson said:
In , the size of the internal hom, , is known to be , however, we category theorists tend to work up to iso.
But after reading on the 12fold way something bothers me about this: does the size of hom-sets not change when thinking up to iso?
Nope! That's the exact reason that we can think of isomorphic objects in a category as being "the same": the morphisms from or to any other object transfer (bijectively) from an object to any isomorphic one by composition with the isomorphism.
Kale Evans (Aug 09 2021 at 15:13):Sincerely thank you for all of your help Prof @John Baez . I learned a lot of new things.
John Baez (Aug 09 2021 at 18:00):Sure!
Kale Evans (Aug 13 2021 at 05:21):Quick question. What is the difference between a function that maps to the underlying set of a Monoid, vs a function that maps to the Monoid?
I've seen both used in the literature. Does the latter just contain the identity element in addition to the underlying set, as well as objects resulting from ?
Spencer Breiner (Aug 13 2021 at 13:06):Properly speaking "a function that maps to the Monoid" is an abuse of terminology, as functions live in and monoids live in . It would be more precise to say "a function that maps to the underlying set of the monoid", but we often conflate and .
The free monoid construction ( with concatenation as binary op) lets you represent the same data (a function into the underlying set) as a monoid homomorphism out of the free monoid. I.e., these two arrows contain the same information:
If you haven't met it yet, this general pattern is called an adjunction.
Joe Moeller (Aug 13 2021 at 13:57):A perspective to take here is that "function mapping to a monoid" is something most mathematicians would say and not blink an eye, and "underlying set" is a thing most often said by the sort of mathematician who likes category theory a lot. I've heard many people say that making this distinction is overly pedantic. It's simply the sort of thing that a category theorist benefits from being careful about.
Kale Evans (Aug 13 2021 at 14:56):I see. Yeah, I've read papers that have used both terminologies interchangeably, and was wondering if one was different from the other. It sounds like in both instances, its just a map to the set, and there is not subtle differences to be aware of
Nick Hu (Aug 13 2021 at 15:10):I've seen people say 'function' when they mean 'homomorphism', not frequently, but it can happen
John Baez (Aug 14 2021 at 04:04):Kale Evans said:
I see. Yeah, I've read papers that have used both terminologies interchangeably, and was wondering if one was different from the other. It sounds like in both instances, its just a map to the set, and there is not subtle differences to be aware of.
I think everyone here was trying to tell you there's no difference, but for some reason they didn't want to just come out and say just that.... probably because it's kind of interesting.
Fawzi Hreiki (Aug 14 2021 at 06:50):For something like monoids there isn’t really a difference but it becomes more important when you deal with structured object which have multiple layers (e.g. schemes which have both underlying topological spaces and underlying sets)
Fawzi Hreiki (Aug 14 2021 at 06:52):Or even more interestingly, something like topological or Lie groups since sometimes you want to talk about just continuous/smooth maps which are not necessarily group homomorphisms and vice versa
Kale Evans (Dec 14 2021 at 23:17):I feel I'm the only one asking questions in this space:big_smile:. Well, I suppose I'm ok with that...
Question. Can anyone provide some intuition with regards to how categorical limits and colimits generalize the idea of intersection and disjoint union in sets? I've heard this as being the case, but having some trouble finding an explanation for this.
John Baez (Dec 15 2021 at 00:37):Intersections and unions are products and coproducts, respectively - but not in the category of sets! Rather, they are products and coproducts in the category of subsets of a given set. For any set there's a category where the objects are subsets and there's one morphism from to if , and none otherwise.
John Baez (Dec 15 2021 at 00:38):It would be a very good exercise to check that intersections and unions are product and coproducts in this category. Then this fact can be generalized in various ways.
Patrick Nicodemus (Dec 15 2021 at 01:53):Kale Evans said:
I feel I'm the only one asking questions in this space:big_smile:. Well, I suppose I'm ok with that...
Question. Can anyone provide some intuition with regards to how categorical limits and colimits generalize the idea of intersection and disjoint union in sets? I've heard this as being the case, but having some trouble finding an explanation for this.
let be a set and be two subsets of . Let be the two inclusion maps of respectively into . The pullback square associated to has in the top left corner. (A pullback is a special case of a categorical limit)
If are two sets, then the disjoint union of and is their categorical coproduct, which is a special type of colimit.
Both of these theorems are proved in any intro category theory textbook. (Or posed as an exercise to the reader.)
There's some abuse of language here in that a colimit/limit is really a diagram with arrows and objects, and I am just talking about the objects here.
David Egolf (Dec 15 2021 at 02:09):Kale Evans said:
Question. Can anyone provide some intuition with regards to how categorical limits and colimits generalize the idea of intersection and disjoint union in sets? I've heard this as being the case, but having some trouble finding an explanation for this.
In terms of intuition, I think of limits and colimits as being "extreme" or "optimum" with respect to some property. That is closely related to the the way I think about the existence of a unique morphism to (or from) these special objects.
To take your example of the intersection - If we have two sets of interest and , and we want to consider sets that are subsets of both, then the "biggest" such set (the one containing as many elements as possible) is the intersection of and .
The union is similar. Again we have two sets of interest and , but now we want to consider sets that contain both and as subsets. The "smallest" such set (the one containing the minimum number of elements possible) is the (disjoint?) union.
Disclaimer: I'm just learning how all these things work myself!
Fawzi Hreiki (Dec 15 2021 at 10:16):An intuition I like to keep in mind (which isn’t always accurate) is that limits are algebraic and they ‘cut’ or ‘solve’ while colimits are geometric and they ‘glue’
Fawzi Hreiki (Dec 15 2021 at 10:16):The limit is the ‘largest possible solution’ while the colimit is the ‘smallest possible glueing’.
Fawzi Hreiki (Dec 15 2021 at 10:17):At the end of the day though, limits and colimits are extremely general and formal notions. So the only way you’ll get any intuition is just by looking at a lot of very different examples.
Peiyuan Zhu (Dec 15 2021 at 19:50):What's the difference between an 'arrow' and a function in a category?
Fabrizio Genovese (Dec 15 2021 at 19:51):A function is an arrow in the category of sets and functions
Fabrizio Genovese (Dec 15 2021 at 19:51):While an arrow is just another word for "morphism"
Peiyuan Zhu (Dec 15 2021 at 19:56):In the definition of a category, there's no mention of any particular structure, yet the word 'morphism' is used. What kind of structure is required or is it just saying one defines category at a generality independent of structures?
Peiyuan Zhu (Dec 15 2021 at 19:56):How about the difference between an 'object' and a set?
Peiyuan Zhu (Dec 15 2021 at 19:59):Fabrizio Genovese said:
A function is an arrow in the category of sets and functions
Are relations arrows?
John Baez (Dec 15 2021 at 20:21):Peiyuan Zhu said:
In the definition of a category, there's no mention of any particular structure, yet the word 'morphism' is used. What kind of structure is required or is it just saying one defines category at a generality independent of structures?
A category has things called "morphisms" and some things called "objects", but they can be absolutely anything - we just need some name for them, nothing is required besides what's actually in the definition of category
Joe Moeller (Dec 15 2021 at 20:21):You should look at a gazillion example of categories to get the difference between functions and arrows and objects and sets. There are categories that literally are collections of sets with extra structure:
etc etc. But then there are also weirder examples of categories:
I think once you get a hold of why all these can all be called category, your questions will be answered.
Fabrizio Genovese (Dec 15 2021 at 21:15):Joe Moeller said:
You should look at a gazillion example of categories to get the difference between functions and arrows and objects and sets. There are categories that literally are collections of sets with extra structure:
- the category of sets and functions,
- the category of groups and homomorphisms,
- the category of toplogical spaces and continuous maps,
etc etc. But then there are also weirder examples of categories:
- a set is a category where the elements are the objects, and there's only identity morphisms,
- a partially ordered set is a category where the objects are the elements and the morphisms are just the relation of one being less than another,
- a group is a category where there is one object (there's nothing at all to this object, it's essentially a placeholder) and the morphisms are the elements of the group,
- if X is a space, then there's a category where the objects are points in the space and a morphism is a path from one point to another
I think once you get a hold of why all these can all be called category, your questions will be answered.
Yes, and to add to this, relations are arrows in the category where sets are objects and relations are morphisms. But you could also define a notion of "morphism between relations" and end up with a category where relations play the role of objects. This may seem confusing, but keep in mind that the most important holistic principle in category theory is
everything is everything else!
Fabrizio Genovese (Dec 15 2021 at 21:17):That is, often what you really do in category theory is describing the same thing from different perspectives. For instance, you could describe limits/colimits as terminal/initial cones over a diagram, or as adjunctions. The things you are describing stay the same, what changes is the perspective from which you describe them.
Fabrizio Genovese (Dec 15 2021 at 21:19):Tools like limits, colimits, adjunctions, Kan extensions and others have indeed this strange property that you can basically define one using the other, and vice-versa. If you think about a concept as a cube, then defining the concept using one or another tool amounts to look at the cube from different perspectives, e.g. by looking at different faces.
Peiyuan Zhu (Dec 15 2021 at 23:38):But if 'everything is everything else', then nothing is not anything else. Given a statement, how do you determine whether a statement is true? Given a definition, how do you determine whether it is self-consistent? For example, in classical logic I can check each and every term to make sure they're well-defined and there's no contradiction; the truth of a theorem is determined by the truth table assuming the axioms or definitions. But as you said these are illegitimate here.
Peiyuan Zhu (Dec 16 2021 at 00:00):Is there an example from information geometry about what is a functor? While reading Čencov's 'Statistical Decision Rules and Optimal Inference', this concept appears several times in the introduction chapter (P57~59) and seems to have something to do with sufficient statistics later on (P75 Lem 5.12; P95 Lem 7.4) and appear with features such as 'covariant', 'contravariant', 'heredity invariant', and 'monotone invariant'. Nowhere did it mention 'adjointness' and 'Galois connection' but they seems to be relevant in this context.
Peiyuan Zhu (Dec 16 2021 at 00:13):Attached are the lemmas and definitions:
image.png
image.png
image.png
Peiyuan Zhu (Dec 16 2021 at 00:19):It seems like a functor maps between categories, acting on arrows as well as objects. It is again not necessarily a function just like arrows are not necessarily functions. It is the compositionality that matters, rather than ontological existence.
Fawzi Hreiki (Dec 16 2021 at 00:44):The term function is a bit ambiguous. It can mean a map of sets, but it can also mean a special kind of map in geometry, for example a real valued function on a space.
Fawzi Hreiki (Dec 16 2021 at 00:45):Or a regular function on an algebraic variety.
Peiyuan Zhu (Dec 16 2021 at 01:41):I think an important thing for me to know is what is the right questions to ask in category theory, which seems quite different from classical logic
John Baez (Dec 16 2021 at 02:30):Peiyuan Zhu said:
But if 'everything is everything else', then nothing is not anything else. Given a statement, how do you determine whether a statement is true? Given a definition, how do you determine whether it is self-consistent? For example, in classical logic I can check each and every term to make sure they're well-defined and there's no contradiction; the truth of a theorem is determined by the truth table assuming the axioms or definitions. But as you said these are illegitimate here.
No - all the methods of standard mathematics are completely legitimate here. Definitions and theorems work in category theory just like they do in any other branch of math - until you get to advanced stuff where you use category theory to come up with new rules of logic.
Fabrizio was making some advanced "philosophical" points, but you are clearly not ready for those yet, so they are just confusing to you.
John Baez (Dec 16 2021 at 02:31):After a few years they may make sense, but right now it's probably good for you to ignore them. I think the important thing is to learn lots of basic definitions, examples, theorems, and proofs.
John Baez (Dec 16 2021 at 02:34):You should probably start by learning about 20 very different examples of categories, 20 examples of functors between them, and 20 examples of natural transformations between those.
Fabrizio Genovese (Dec 16 2021 at 12:53):I agree with John, you are probably not ready. But just to give a simple example, you may say that
"A (small) category consists of a set of objects, a set of morphisms, and some laws and axioms that must hold."
In this case, you are considering the theory of sets as primitive, start from there and say what categories are. But there are also ways to axiomatize what categories are without resorting to set theory. Ad that point, you may say that
"A set is a (small) category where there are only identity arrows. A function between set is a functor between such categories."
I put the "small" thing in parentheses because it's not important to focus on it right now.
...Here you are using category theory as primitive and defining set theory using that. Who comes first? Well, it depends on your point of view! The point is exactly that many concepts in category theory can be defined in multiple ways, that can be proved equivalent. Once you pick one, you can use it to define other concepts, or vice versa, a bit like in the example above
Fabrizio Genovese (Dec 16 2021 at 12:55):You mentioned classical logic, and well, in classical logic things don't come out of nowhere. You chose some axioms and start building stuff from there. Those axioms are your "primitive facts". You could choose different sets of axioms and get equivalent theories out of them. That is, the same theory can be axiomatized in different ways. Something along those lines is happening here :smile:
Mike Shulman (Dec 16 2021 at 14:32):However, no one has ever really given a convincing set of axioms for category theory that would enable you to define a set as a discrete small category and develop all of mathematics from there.
Peiyuan Zhu (Dec 16 2021 at 21:05):I see. So I can accept that there’s no ‘definition’ for objects and morphisms, and in case there’s an identity and commutativity that thing is a category. What is not a category is for example a circular order defined on a set. Here the morphisms are orders and they do not commute. Taking identity operator away from any set makes it not a category. For example, the linear transformations between vector spaces as morphisms on objects. Taking the identity matrix away makes it less than a category.
Peiyuan Zhu (Dec 16 2021 at 21:19):How do you understand the univalence axiom? What’s the difference between intentionality and extensionality? How does this relate to the 'structrualism' vs 'object realism' debate?
David Egolf (Dec 16 2021 at 23:32):Peiyuan Zhu said:
I see. So I can accept that there’s no ‘definition’ for objects and morphisms, and in case there’s an identity and commutativity that thing is a category. What is not a category is for example a circular order defined on a set. Here the morphisms are orders and they do not commute.
I'm not quite following your line of thought, but for what it's worth morphisms often don't commute in categories. For example, any group can be viewed as a category with a single object, where the object is some set (say ) and the morphisms are the elements of the group. If the group isn't commutative, then that means that morphisms don't commute when we compose them.
Mike Shulman (Dec 16 2021 at 23:54):Peiyuan Zhu said:
How do you understand the univalence axiom? What’s the difference between intentionality and extensionality?
Those aren't really questions about category theory, but about foundations of mathematics, although category theory is relevant to understanding univalence. In category theory, two objects of the same category that are isomorphic in that category have all the same "categorical properties". E.g. two isomorphic groups have all the same group-theoretical properties, so that group theorists speak of "the cyclic group of order 3" because such a group is unique up to isomorphism, even though strictly speaking there are many of them.
The univalence axiom is an addition to the foundations of mathematics (which doesn't make sense if added to a ZFC-style foundation, but does make sense if added to a type-theoretic foundation) that eliminates the difference between equality and isomorphism for objects of such categories. Under univalence, two isomorphic groups are in fact "equal", although this can be a misleading thing to hear at first because it arises by expanding the notion of "equal" so that it means "isomorphic", rather than somehow "squashing" or "quotienting" to force isomorphic groups to become equal in the original sense of "equal". In particular, just as two groups can be isomorphic in more than one way, under univalence two groups can be equal in more than one way.
Mike Shulman (Dec 17 2021 at 00:02):For the second question, we say that a statement or property is intenSional (not intenTional) if it depends on how things are defined, and extensional if it depends only on what those things are. For instance, the morning star and the evening star are intensionally different, but extensionally equal (both being the planet Venus). Mathematically, "" and "the number of exotic 7-spheres" are intensionally different, but extensionally equal.
You may have heard these words used to describe type-theoretic foundations of mathematics. In this case, for historical reasons, their use is extremely confusing and often almost "backwards" from what it ought to be. In type-theoretic foundations, there are (at least) two different kinds of "equality": a "judgmental" equality and a "typal" (or, more traditionally, "propositional") equality. The typal equality is the "real" equality of mathematics, which is usually extensional: "" and "the number of exotic 7-spheres" are typally equal. (It's also the equality that is expanded by the univalence axiom to mean "isomorphism".) To be sure, sometimes one considers type theories in which the typal equality is not fully determined, e.g. function extensionality may fail, so that it is not as extensional as it might be; but it is always at least somewhat extensional.
The judgmental equality is potentially distinct from the typal equality, and thus potentially intensional. One says that an "intensional type theory" is one in which the judgmental equality is truly intensional, whereas an "extensional type theory" is one in which the judgmental and typal equality coincide and are thus both extensional. Unfortunately, this sometimes leads people to apply these adjectives to the typal equality itself and speak of an "intensional equality type" or an "extensional equality type", whereas in fact as I said the typal equality is always extensional (or at least aspires to be so) and the adjective really refers to the behavior of the judgmental equality.
Ironically, the univalence axiom, which is an ultimate sort of extensionality property for the typal equality, is inconsistent with extensional type theory. Thus, it is intensional type theory that can actually be "more extensional", at least in the behavior of its typal equality.
Mike Shulman (Dec 17 2021 at 00:03):That's probably way more than you wanted to know! But there's a lot of confusion around these things, so I wanted to clarify.
Peiyuan Zhu (Dec 28 2021 at 19:52):[1] Why is matrix multiplication instead of ? Why is the former has some consistency with logical operations but not the latter when think of it in terms of composition of relations? e.g. , What does this have to do with the notion of 'semiring' and what's useful about this algebraic notion?
[2] Does Hom(-,-) mean a set so it has no multiplicities? In this case, it seems like the set theory is useful? (I thought I could just throw set theory into the garbage can) What is the point of defining 'partial functions'?
[3] When people say 'category theory is relational' they're talking about a properties of the category theory, or are they talking about 'category of relations', or is this distinction important?
Fawzi Hreiki (Dec 28 2021 at 20:21):Hint: composition of linear maps
John Baez (Dec 28 2021 at 20:54):[2] In a category Hom(-,-) is a set. Set theory is very useful in category theory, just like it is in most mathematics. When you learn category theory it's like learning other mathematics: you don't throw previous mathematics that you learned into the garbage can!
(Again, experts may be tempted to argue with me here, since there are alternatives to set theory, but this is a beginner question, and a beginner should not be shocked that category theory textbooks use sets a lot.)
The point of defining partial functions is that lots of functions are partial. Think of functions like , and so on. They are partial functions from to .
Simon Burton (Dec 29 2021 at 16:40):[1] You have reminded me that John Baez has a nice video and/or notes discussing how linear algebra works over a semi-ring, and you can do path minimization (like Dijkstra's algorithm) when using the min-sum semiring. There's also this blog https://graphicallinearalgebra.net/ which you might find interesting. Anyway, I've been meaning to ask @John Baez where to find these notes of his...
John Baez (Dec 29 2021 at 20:01):Hmm, I'm not sure I know what you're talking about, but maybe you mean weeks 11, 12 and 13 here:
Nathaniel Virgo (Dec 30 2021 at 03:46):I recently watched a really nice talk by Jade Master on those topics
Peiyuan Zhu (Jan 01 2022 at 07:36):Very often I see notations Hom(-,X) where "-" is used. What does that mean? An argument by seeing Hom as a "function" by fixing the second argument as object X?
Peiyuan Zhu (Jan 01 2022 at 08:03):What does mean in the diagrams? What is it the diagram trying to say? is the characteristic function that takes value 1 if the argument is in the set and zero otherwise. image.png
Peiyuan Zhu (Jan 01 2022 at 08:12):I'm finishing chapter 7 of the online course on category theory, very interesting! It seems that here we're making a lot of constructions in many different mathematical fields such as linear algebra and differential geometry very explicit, in a very succinct and insightful way. I wonder if there's such a statement as 'use category theory to prove something', or is it just showing the structures? If it can 'prove something', what are the 'legitimate moves'? What is an acceptable writing or do people accept 'proof by diagram'?
Peiyuan Zhu (Jan 01 2022 at 08:26):It's very satisfying that the course uses mostly examples from linear algebra, which is a very good review from a brand new perspective, straight to the key ideas that appears in all sorts of places.
Peiyuan Zhu (Jan 01 2022 at 08:30):Where do the numbers 60,15,105,5 come from in this image? is the action of multiplying the argument by . What is this proof trying to say? image.png
Peiyuan Zhu (Jan 01 2022 at 19:30):Fawzi Hreiki said:
Hint: composition of linear maps
Is there a 'relational view' of this? A special kind of matrix can be used to describe relations, and real matrices are weighted, signed relations.
Morgan Rogers (he/him) (Jan 01 2022 at 23:08):@Peiyuan Zhu that's a lot of questions! I'll answer a few and hope others fill in the rest.
Hom(-,X) does indeed have the meaning you deduced; X is fixed, while the blank entry is intended to be filled by the argument of the functor (with a morphism being mapped to the function "compose with ").
A double arrow, , in a diagram indicates a 2-morphism. That is: there are two ways around the diagram, representing different composite 1-morphisms. In a commutative diagram, these composites are supposed to be equal, but there are situations where instead there is merely a comparison 2-morphisms in one direction or the other.
For example, if the 1-morphisms are functors, the arrow indicates a natural transformation between the composite functors on either side of the relevant cell of the diagram.
Jules Hedges (Jan 05 2022 at 11:07):Mike Shulman said:
I find it very confusing to have a single topic called "beginner questions" in which lots of very different questions and their answers get mixed together. Wouldn't it make more sense to open a separate topic for each question?
[moderator hat] Yes, please do!
Morgan Rogers (he/him) (Jan 05 2022 at 11:07):Mike Shulman said:
I find it very confusing to have a single topic called "beginner questions" in which lots of very different questions and their answers get mixed together. Wouldn't it make more sense to open a separate topic for each question?
In principle moderators can branch topics, but I haven't personally had ore than phone access until now. I'll arrange things...
Peiyuan Zhu (Jan 06 2022 at 03:20):Did people use topos theory in proof assistants? Does modality give rise to new features of theorem proving?
Jon Sterling (Jan 06 2022 at 12:27):Yes! My colleagues and I have recently used open and closed modalities from topos theory to axiomatize and prove things about the cost/complexity of computer programs. We have implemented this in the Agda proof assistant.
Peiyuan Zhu (Jan 06 2022 at 17:16):What does ‘internal logic’ or ‘internalize’ mean? http://nlab-pages.s3.us-east-2.amazonaws.com/nlab/show/internal%20logic I guess this is essential to understand what a groupoid is? (as a motivation)
Peiyuan Zhu (Jan 06 2022 at 18:02):Is there any reading anyone recommend around the lines of ‘algebra of concepts’?
Peiyuan Zhu (Jan 06 2022 at 18:02):209BC638-F373-4F45-AC59-E90C52C7F4EA.png
Peiyuan Zhu (Jan 06 2022 at 18:02):067E1E2E-70A1-435C-B38C-049C9A4A9CA7.png
Peiyuan Zhu (Jan 06 2022 at 18:03):Also what kind of mathematical tools are suitable to study these algebraic structures?
Peiyuan Zhu (Jan 06 2022 at 18:06):To what extend is category theory relevant here?
Peiyuan Zhu (Jan 06 2022 at 18:38):I think it has something to do with judgemental/definitional equality in HoTT, but still trying to see exactly how
Peiyuan Zhu (Jan 06 2022 at 20:03):Does Topos/category theory has anything to do with lattice/Galois theory?
John Baez (Jan 07 2022 at 00:39):What does ‘internal logic’ or ‘internalize’ mean?
Ordinary mathematics is done in the category using ordinary logic. Generalizing from this, people realized that every category has its own form of logic, suitable for working within that category. This is called the 'internal logic' of that category.
This idea is very broad and general, but topoi are categories rather similar to , so their internal logic is rather similar to the logic you know from school. The internal logic of a topoi is a well-studied subject, and when you're done reading Sheaves in Geometry and Logic you'll know a fair amount about it: that book is a pretty good introduction.
I guess this is essential to understand what a groupoid is? (as a motivation)
No, a groupoid is just a category where all morphisms have inverses. A groupoid with one object is essentially the same thing as a group. So, if you have some way of understanding groups - and there are many - there's a decent chance you can generalize it and understand groupoids.
John Baez (Jan 07 2022 at 00:39):One could say that groupoids are a modern way of understanding symmetry. You may be more familiar with the special case of groups, which are an older way of understanding symmetry.
Peiyuan Zhu (Jan 08 2022 at 04:46):Thanks so much!! I think I have an idea of what's going on
Peiyuan Zhu (Jan 08 2022 at 18:39):Fabrizio Genovese said:
I agree with John, you are probably not ready. But just to give a simple example, you may say that
"A (small) category consists of a set of objects, a set of morphisms, and some laws and axioms that must hold."
In this case, you are considering the theory of sets as primitive, start from there and say what categories are. But there are also ways to axiomatize what categories are without resorting to set theory. Ad that point, you may say that
"A set is a (small) category where there are only identity arrows. A function between set is a functor between such categories."
I put the "small" thing in parentheses because it's not important to focus on it right now.
...Here you are using category theory as primitive and defining set theory using that. Who comes first? Well, it depends on your point of view! The point is exactly that many concepts in category theory can be defined in multiple ways, that can be proved equivalent. Once you pick one, you can use it to define other concepts, or vice versa, a bit like in the example above
Quite an interesting read: When are two objects equal? https://people.math.harvard.edu/~mazur/preprints/when_is_one.pdf Making @Fabrizio Genovese's answer more precise.
Peiyuan Zhu (Jan 08 2022 at 19:22):I don't know if there'd be people calling this 'mystical', I definitely have heard someone saying that in one video that I saw, which I think is facing a group of audience trained in classical logic
Peiyuan Zhu (Jan 08 2022 at 19:23):The book writes "A category is a B.Y.O.S.T. party, i.e., you bring your own set theory to it.". That's really funny.
Peiyuan Zhu (Jan 08 2022 at 19:37):"One could, after all, simply require that there be a set of objects of the category, rather than bring in the airy word class." So it's more rigorously a 'class of objects' not a 'set of objects'.
Peiyuan Zhu (Jan 08 2022 at 20:58):Here I'm gonna go through a simple proof in category theory that says initial objects are unique. It's really fun to read his writing and he made it very precise and concise when building an example like this. I think what I'm doing here is like a 'categorical automata' that does category theory. First there are possible worlds that define a category as an objects with some properties, which includes a truth table, that the first row with a category and three properties: has arrow, has object, has arrows, has identity arrows. There's also equality, truth, false, and implications consisting of what's necessary for a truth table deductive inference, and proof techniques (are the proof techniques monads?) representing possible transformations from objects to objects. The automata is like a Turing machine (not sure if it fit into the definition of Turing machine exactly? I'm not a computer scientist, yet) going through objects and their properties and adding new arrows and erasing arrows (isomorphism, out of memory) and adding new properties (uniqueness) and erasing old properties (make it a category), where arrows indicates possible transformation of objects. The automata intuitively make the hypothesis that suppose there's something else (a distinction to be made here is that this is a proof by contradiction not a proof by negation, so hypothetical objects are okay, similar to proof by negation this is a 'proof by inverse' or 'proof by drawing arrows', the purpose of this is to find a 'hypothetical mapping', I suppose this has something to do with 'monads'?), if this is drawn on a paper it would be a branched out arrow pointing away from that new hypothetical object, and adding one tentative object on that table of categories, creating a new category and create a new possible mapping from that category, and commutativity of two arrows collapses the new object into one, making a new property of uniqueness of that initial object, and since it'a category one add this new object named as 'initial object'.
Peiyuan Zhu (Jan 08 2022 at 21:10):In what way I should make this more rigorous? Would anybody mind asking me a few more questions to make it (the proof machine of category theory) a solid category theoretic proof? As a beginner am blinded by the number of questions that I think I can ask. What shall I call this called in philosophy? Modal realism?
John Baez (Jan 08 2022 at 23:40):Your proof is long and it has many digressions of a philosophical or logical nature, which seem very distracting to me. I also don't see the actual argument that proves the theorem!
John Baez (Jan 08 2022 at 23:40):Here is a more typical proof that initial objects are unique up to unique isomorphism:
John Baez (Jan 08 2022 at 23:41):This is nice and short. It's a bit too short because it doesn't prove is an isomorphism. Doing that would take two extra lines, and adding those would be very useful.
Todd Trimble (Jan 08 2022 at 23:50):The diagram shows are inverse to one another, so they are isomorphisms.
John Baez (Jan 08 2022 at 23:52):Wow, I didn't see any diagram or any morphism .
John Baez (Jan 08 2022 at 23:53):Oh - I didn't even scroll down far enough to see the proof! What I thought was the proof was just part of their statement of the theorem. :blushing:
John Baez (Jan 08 2022 at 23:56):So yes, their proof is complete and fine. Personally I'd do a proof with more words and no diagram, but it would amount to the same thing - it's just a matter of stylistic preference. I prefer a "narrative" when it's not too lengthy. But it's definitely important to see how you can prove things with diagrams in category theory.
John Baez (Jan 08 2022 at 23:56):(As you can see I'm a bit loquacious; my commentary on their proof is longer than their proof.)
Peiyuan Zhu (Jan 09 2022 at 17:18):Boole’s Algebra is not Boolean Algebra: https://www.jstor.org/stable/pdf/2689628.pdf?casa_token=HenX3qBwxp0AAAAA:TvinWrOjbPs8d0lJKU0HSWfhT79CgQ-hl2xjsLBOC1u_Hdaj-IHY3ueDfQmhG7U2xRhO3OjEVHWtEW5CWZrPL6KPIWwdwc3Y1n7_yLSP11vcgPfxqfI
Peiyuan Zhu (Jan 09 2022 at 17:21):I wonder why Algebra seems never has been an issue discussed in the foundation of mathematics, or is it just my ignorance? https://plato.stanford.edu/entries/algebra-logic-tradition/
John Baez (Jan 09 2022 at 18:14):The reference you just gave to that Stanford Encyclopedia of Philosophy article is all about algebra in the foundations of mathematics, or at least algebra in logic. I consider logic to be part of the foundations of mathematics - maybe you don't?
Peiyuan Zhu (Jan 09 2022 at 19:31):I recently think algebra is really important because I just realized in my university math course on proofs we just went through set theory, truth tables, quantifiers, but without seeing the algebraic side of things. And then we went onto group theory but without seeing any logic side of things. My question to myself is whether there can be a unifying picture between the two, and what can be a good entry to these materials?
And then homotopy comes in started with propositions and proofs in analogy with spaces and paths, but I've yet to see a discussion on algebra.
Peiyuan Zhu (Jan 09 2022 at 19:35):I guess I'm hoping to read something more accessible, likely the more in modern language of algebra and reference to undergraduate logic the better. I tried reading Boole, Peirce, Tarski and the language seems very inaccessible to my level. I was trying to see if there're people that I can reach out to but the person named Theodore Hailperin who tried to rewrite Boole in modern language seems to have passed away already.
Peiyuan Zhu (Jan 09 2022 at 19:41):John Baez said:
The reference you just gave to that Stanford Encyclopedia of Philosophy article is all about algebra in the foundations of mathematics, or at least algebra in logic. I consider logic to be part of the foundations of mathematics - maybe you don't?
Do you mean you think algebra less foundational than logic?
John Baez (Jan 09 2022 at 19:46):I don't really care much what's "foundational", personally. So when people say "foundational" I just assume they're using the word in the usual way mathematicians do.
You said "I wonder why Algebra seems never has been an issue discussed in the foundation of mathematics", and then you pointed us to a web page full of discussions of algebra in the foundation of mathematics.
Peiyuan Zhu (Jan 09 2022 at 19:48):John Baez said:
I don't really care much what's "foundational", personally. So when people say "foundational" I assume they're using the word in the usual way mathematicians do.
You said "I wonder why Algebra seems never has been an issue discussed in the foundation of mathematics", and then you pointed us to a web page full of discussions of algebra in the foundation of mathematics... at least according to many people's concept of "foundational".
Yea it's my first time seeing that page. I think I'm missing a lot of things.
Peiyuan Zhu (Jan 09 2022 at 19:50):I guess I should've been more precise in saying 'the foundation that I thought was the foundation'
John Baez (Jan 09 2022 at 19:51):If you're trying to learn about the connection between algebra and logic, there's tons to read about that. Along with the Stanford Encyclopedia of Philosophy page, which focuses on a lot of older work, these pages give lots of references:
Wikipedia, Algebraic logic.
Wikipedia, Abstract algebraic logic.
Peiyuan Zhu (Jan 09 2022 at 19:53):Interesting, I'm curious to know why this is not in a university math degree curriculum anymore... I was graduated with an honours degree in a North American university but was never told there's something like this.
John Baez (Jan 09 2022 at 19:55):I bet I could list dozens of subjects you've never heard about.
Peiyuan Zhu (Jan 09 2022 at 19:55):Also didn't remember anything like this in the computer science department too
Peiyuan Zhu (Jan 09 2022 at 19:56):John Baez said:
I bet I could list dozens of subjects you've never heard about.
Sure but it think it would've made an interesting ugrad course
John Baez (Jan 09 2022 at 19:58):That's also true of dozens of other subjects. Did you see a course on knot theory, a course on projective geometry, a course on the lambda-calculus, a course on quaternions, a course on elliptic functions, or a course on the basics of p-adic analysis...? These are all lots of fun and could make great undergrad courses, but most universities don't teach them because they don't have any faculty eager to teach them.
John Baez (Jan 09 2022 at 19:59):Basically the problem is that there's a lot of math to learn.
Peiyuan Zhu (Jan 09 2022 at 21:05):Interesting read on Whitehead's Universal Algebra: https://www.religion-online.org/article/whiteheads-early-philosophy-of-mathematics/
John Baez (Jan 09 2022 at 21:14):Whitehead's universal algebra sucked compared to Birkhoff's universal algebra, which later gave rise to the study of Lawvere theories.
John Baez (Jan 09 2022 at 21:18):Unlike Whitehead, Birkhoff proved powerful general theorems about universal algebra. But we can at least credit Whitehead for getting people excited about the idea of universal algebra.
Peiyuan Zhu (Jan 10 2022 at 03:47):I don't really understand these two paper but it seems to be something important that I need to know, which mentions algebraic logic and universal algebraic geometry. [1] https://arxiv.org/pdf/1205.6235.pdf [2] https://link.springer.com/content/pdf/10.1007%2F3-540-44669-9_5.pdf I'm reading this paper together with information algebra with respect to universal algebra: https://www.mdpi.com/2078-2489/5/2/219/htm To quote a few sentences from geometry of uncertainty "As mentioned above, we start by observing that Dempster’s rule can be easily extended to normalised sum functions. This is necessary since, as we will see later on, the geometry of the orthogonal sum can only be appropriately described in terms of whole affine spaces which do not fit within the confines of the belief space, a simplex." "Strikingly, for any subset A, the resulting affine spaces have a common intersection
for all k 2 [0, 1], which is therefore characteristic of Bel.". This is interesting because just at terminology-level I can see sensible correspondence.
Peiyuan Zhu (Jan 10 2022 at 05:17):image.png
"We will consider also a situation when the system T contains not only equalities, but arbitrary formulas of FOL. In other words, we describe knowledge in some logical language and interpret its content as a geometrical image.", I understand this as saying geometry is necessary for giving semantics of first-order-logics. Is there a reason for this?
John Baez (Jan 10 2022 at 05:20):That paper looks too much like philosophy for me to understand it - stuff about "the subject of knowledge", etc.
Peiyuan Zhu (Jan 10 2022 at 08:56):I think it’s pretty consistent with what David Hestenes was doing. Not sure if you have ever encountered his ‘geometric algebra’. Here’s an interview of him http://worrydream.com/refs/Hestenes%20-%20An%20Interview%20with%20David%20Hestenes.pdf
Jon Sterling (Jan 10 2022 at 09:21):@Peiyuan Zhu With respect, I think that your development in your areas of interest will be greatly hastened if you spend some time trying to read just one thing from one angle on one topic. I am noticing that you are interested in a lot of very diverse things, which is very good, but I think this approach is not likely to lead to achieving (what I think are) your goals. I am also sensing that people here are starting to experience whiplash from the rapid turns from topic to topic.
I hope that you find this message encouraging rather than discouraging, and I apologize if I have phrased it poorly. I think that if you narrow down your goals to something specific, people in this channel will be able to provide you assistance in mastering it. If this advice isn't helpful to you, please feel free to ignore it.
Peiyuan Zhu (Jan 10 2022 at 09:31):That’s a good advice and I’m spending time on doing small things. The thing I’m trying to understand is semiotics, and this has been my goal. It’s really broad because whenever you’re using sign it immediately become an important part of it. And it’s not a joke that it is important to mathematics, physics, biology, computer science, etc. Of course in the end it has to be narrowed down to a specific field to work on and I’m now in the process of making something manageable and interesting and challenging. I think I’m coming close.
Peiyuan Zhu (Jan 10 2022 at 09:32):But maybe I can open a channel for this topic only, and leave the beginner channel for more CT questions
Peiyuan Zhu (Jan 10 2022 at 09:37):To quote from that interview “All the great quantum physicists, Pauli, Schroedinger, Heisenberg and even Dirac as well as mathematicians Weyl and von Neumann, failed to recognize its geometric meaning and the fact that it has nothings to do with spin. When you see the Pauli sigmas as vectors, then you can see the identity as expressing the simple geometric fact that three orthogonal vectors determine a unit volume. Thus there is geometric reason for the Pauli algebra, and it has nothing whatsoever to do with a spin. When I completed my little lecture on geometry of the Pauli algebra, my father gave me the greatest compliment of my life, which I remember to this day. He said, “You have learned the difference between a mathematical idea and its representation by symbols. Many mathematicians never learn that!” That really made me proud, because, actually, my father was very sparing with his compliments. So when he praised me, I knew I knew I had really done something good.” This is talking about semiotics in physics with geometry as semantics.
Morgan Rogers (he/him) (Jan 10 2022 at 09:41):It's always a red flag to me when someone claims that a tool used by a large number of experts "has nothing to do with" what they're using it for...
@Peiyuan Zhu don't open a new channel, just find a more specific channel that's relevant and open a topic there; there are plenty to choose from :)
John Baez (Jan 10 2022 at 15:46):When you see the Pauli sigmas as vectors, then you can see the identity as expressing the simple geometric fact that three orthogonal vectors determine a unit volume.
Which identity?
David Egolf (Jan 10 2022 at 16:24):Peiyuan Zhu said:
That’s a good advice and I’m spending time on doing small things. The thing I’m trying to understand is semiotics, and this has been my goal.
If you have access to it, you might consider taking a look at "18 Unconventional Essays on the Nature of Mathematics". This includes a chapter called "Towards a Semiotics of Mathematics". I haven't read this, but it sounds relevant.
I found this opening quote interesting:
semiotics intro
It feels to me that "What is the nature of mathematical language?" is one that comes up implicitly in the context of category theory quite a bit. In particular, I think the aim is often to figure out a language that makes mathematical thinking easier. For example, I remember seeing the argument that it's much more visually straightforward to prove some specific things using diagrams in 2D, as compared to using equations in 1D.
Peiyuan Zhu (Jan 10 2022 at 16:28):John Baez said:
When you see the Pauli sigmas as vectors, then you can see the identity as expressing the simple geometric fact that three orthogonal vectors determine a unit volume.
Which identity?
The identity that three matrices multiples to
John Baez (Jan 10 2022 at 17:16):Okay, good - that makes sense.
Peiyuan Zhu (Jan 10 2022 at 17:25):David Egolf said:
Peiyuan Zhu said:
That’s a good advice and I’m spending time on doing small things. The thing I’m trying to understand is semiotics, and this has been my goal.
If you have access to it, you might consider taking a look at "18 Unconventional Essays on the Nature of Mathematics". This includes a chapter called "Towards a Semiotics of Mathematics". I haven't read this, but it sounds relevant.
I found this opening quote interesting:
semiotics introIt feels to me that "What is the nature of mathematical language?" is one that comes up implicitly in the context of category theory quite a bit. In particular, I think the aim is often to figure out a language that makes mathematical thinking easier. For example, I remember seeing the argument that it's much more visually straightforward to prove some specific things using diagrams in 2D, as compared to using equations in 1D.
I think he's advocating the process of iteratively building better algebraic structures by creating new, more succinct notations with deeper insight using the new distinctions drawn from geometry, combining the distinctions from the existing algebraic systems. In that paper, this process creates a Galois connection between the syntax space and the semantics space. But the distinction means that it's fuzzy in nature, mitigated only by new distinctions rather than definitions, as it will not fit into any existing algebraic structures. So I think this is where topos theory comes in. I don't know if topos theory has anything to do with Galois theory yet, but I don't think I'm talking about nonsense here.
Peiyuan Zhu (Jan 10 2022 at 22:35):Interesting read “Logic as a special case of geometry”: https://www.technology.org/2014/10/23/geometry-fundamental-logic/ Maybe the right thing to do with geometry of uncertainty is see if it is possible to take away the existing explanations that are algebraic, keep only geometry and see if the same constructions can follows without them, or any formulation that can provide better insight of the constructions. I think the author has been searching for a good geometric language to explain the theory, where he found some didn’t work like matorid theory, and some are helpful like lattice theory, some terminologies in algebraic geometry were helpful, and some he expects to be helpful like random set theory but I think it maybe too limited.
Peiyuan Zhu (Jan 13 2022 at 22:01):Aha I finally found the connection! Really solid, just writing this up
Peiyuan Zhu (Jan 14 2022 at 06:58):John Baez said:
Whitehead's universal algebra sucked compared to Birkhoff's universal algebra, which later gave rise to the study of Lawvere theories.
Reading this article on universal geometric algebra: interesting that it gives much more credit to Whitehead and Grassmann than Birkhoff: http://geocalc.clas.asu.edu/pdf/UGA.pdf
Zhen Lin Low (Jan 14 2022 at 07:57):I think you're getting misled by similar names again. This appears to be an article more to do with so-called geometric algebra and less to do with universal algebra.
John Baez (Jan 14 2022 at 18:38):Yeah, Hestenes doesn't do anything with universal algebra. He does Clifford algebras in a very pretentious way - first calling them "geometric algebra", and now in this paper "universal geometric algebra".
Peiyuan Zhu (Jan 16 2022 at 06:08):Are lattices the most universal characterization of algebraic systems? Would it be possible to make a claim that every algebraic system is a lattice?
John Baez (Jan 16 2022 at 07:28):No, and no. (I mean, you can claim anything, but this would not be correct.)
Peiyuan Zhu (Jan 16 2022 at 07:44):Is there an example of an algebra that's not a lattice (of course poset without meet or join but that wouldn't be logical so it wouldn't be an algebra isn't it)? I can't intuitively think of group as a lattice but I also don't know if it is impossible
Jon Sterling (Jan 16 2022 at 10:00):What do you mean "wouldn't be logical"? It is true that the notion of a poset is not an algebraic theory in the ordinary sense, but there are many kinds of algebraic theories, and posets fit into the more general picture.
Jon Sterling (Jan 16 2022 at 10:05):Re groups and lattices, there is a representation theorem from the 1940s that states that every lattice is isomorphic to a sublattice of the subgroup lattice of some group. But this doesn't mean that groups and lattices are the same thing, though of course it does mean that one can sometimes state useful properties about one thing in terms of the other, and vice versa.
Fabrizio Genovese (Jan 16 2022 at 15:34):Peiyuan Zhu said:
Are lattices the most universal characterization of algebraic systems? Would it be possible to make a claim that every algebraic system is a lattice?
No, but lattices are a fundamental tool in universal algebra, since many properties of algebraic structures can be studied by looking at the proprerties of the lattices the congruences on a structure form
Fabrizio Genovese (Jan 16 2022 at 15:34):For instance, the lattice of congruences of a lattice is always distributive
Fabrizio Genovese (Jan 16 2022 at 15:34):So lattice theory is a fundamental tool in (classical, non category theory based) universal algebra, for sure
Fabrizio Genovese (Jan 16 2022 at 15:35):I have no idea how people do this in Lawvere theory, I am only familiar with classical universal algebra.
Fabrizio Genovese (Jan 16 2022 at 15:36):Then you can ask more questions, such as "what are all the varieties of algebras (in the sense of Birkhoff) such that their lattices of congruences obey property X (e.g. modularity?)"
Fabrizio Genovese (Jan 16 2022 at 15:36):This is related with Mal'Cev theory and a lot of other stuff that I remember only faintly.
John Baez (Jan 16 2022 at 17:29):Peiyuan Zhu said:
Is there an example of an algebra that's not a lattice? [...] I can't intuitively think of group as a lattice but I also don't know if it is impossible
You can't usually think of a group as a lattice.
You can't usually think of a ring as a lattice.
You can't usually think of a commutative ring as a lattice.
You can't usually think of a vector spaces as a lattice.
You can't usually think of a Lie algebra as a lattice.
And so on.
John Baez (Jan 16 2022 at 17:30):(of course poset without meet or join but that wouldn't be logical so it wouldn't be an algebra isn't it)
You're using the word "logical" in a strange way that I don't understand here.
Peiyuan Zhu (Feb 04 2022 at 18:19):I read this entry today on relationalism. https://en.wikipedia.org/wiki/Relationalism Is there any mathematical ideas that are relevant to relationalism in particular?
Leopold Schlicht (Feb 04 2022 at 18:32):Peiyuan Zhu said:
I read this entry today on relationalism. https://en.wikipedia.org/wiki/Relationalism Is there any mathematical ideas that are relevant to relationalism in particular?
This is a philosophical topic, so maybe it's more helpful to ask this question philosophers rather than a category theory community. :-)
Peiyuan Zhu (Feb 04 2022 at 18:55):Leopold Schlicht said:
Peiyuan Zhu said:
I read this entry today on relationalism. https://en.wikipedia.org/wiki/Relationalism Is there any mathematical ideas that are relevant to relationalism in particular?
This is a philosophical topic, so maybe it's more helpful to ask this question philosophers rather than a category theory community. :-)
I think it's also a physics question. It says "Relationist physicists such as John Baez and Carlo Rovelli have criticised the leading unified theory of gravity and quantum mechanics, string theory, as retaining absolute space. Some prefer a developing theory of gravity, loop quantum gravity, for its 'backgroundlessness'."
John Baez (Feb 04 2022 at 19:47):Yeah, category theory is deeply connected to relationalism: the Yoneda lemma is sometimes seen as way to make the idea of relationalism precise.
Peiyuan Zhu (Feb 04 2022 at 22:26):John Baez said:
Yeah, category theory is deeply connected to relationalism: the Yoneda lemma is sometimes seen as way to make the idea of relationalism precise.
Is this view compatible with realism?
John Baez (Feb 04 2022 at 22:38):It all depends on what you mean by "realism" - but frankly, I don't care enough about that question to discuss it.
John Baez (Feb 04 2022 at 22:52):Philosophers sometimes throw around "isms" as if they had well-defined meanings, saying stuff like "is relationalism compatible with realism?" It's way too soft and mushy for me. I like math because you can address questions more precisely.
Peiyuan Zhu (Feb 04 2022 at 23:02):I just happened to know that there has been debate in physics on locality so that "local realism" became a synonym for the view held by the authors of the paper of EPR paradox
John Baez (Feb 04 2022 at 23:10):Yeah, and "realism" means about 5 other things too.
Peiyuan Zhu (Feb 04 2022 at 23:11):Ah I see what you mean (e.g. art, politics), but I'm only interested in principle of locality in physics
John Baez (Feb 04 2022 at 23:13):Okay. Local realism is one thing, but realism in the philosophy of mathematics means something else.
Peiyuan Zhu (Feb 04 2022 at 23:16):John Baez said:
Okay. Local realism is one thing, but realism in the philosophy of mathematics means something else.
Are you referring to the realism as an opposition to intuitionism? No I'm not interested in that, obviously I'm in a category theory forum :)
John Baez (Feb 04 2022 at 23:24):Here's an article that tries to explain realism in mathematics:
I really don't care.
Peiyuan Zhu (Feb 04 2022 at 23:34):John Baez said:
Here's an article that tries to explain realism in mathematics:
I really don't care.
Sure, I see what you mean. I only mean whether relationalism in physics is compatible with local realism. If nothing says they're incompatible then they might be, or people might find a way to model it, I guess?
John Baez (Feb 04 2022 at 23:51):Sorry, I really don't care about this either.
Peiyuan Zhu (Feb 05 2022 at 06:24):What is 'principal' about principal fibre bundle? What's interesting about it in applications? Is there a good introductory material to understand this object better? Maybe a book? It looks like the definition of a sheaf, so is it actually a sheaf? Why is this saying it is dependent sum & products "under the guise"? image.png
John Baez (Feb 05 2022 at 06:27):A good intro to principal bundles for someone interested in physics is Analysis, Manifolds and Physics by Choquet-Bruhat, Dillard-Bleick and DeWitt-Morette.
Peiyuan Zhu (Feb 10 2022 at 22:16):There's this nice blog post about groupoid https://math.ucr.edu/home/baez/week247.html that looks very interesting to me. It started from functions from a set to itself to relations from a set to itself. Each function can be identified with a matrix with a single one in each column, and each relation can be identified with a binary matrix. Then it says "We're thinking of them as matrices of numbers. Unfortunately we're still far from getting all matrices of numbers!". Why are we still "far from" getting all matrices of numbers? If we fuzzify the relations to integer degrees of memberships don't we get all matrices of numbers? What is the rationale here?
John Baez (Feb 10 2022 at 22:20):Not all real numbers are natural numbers.
Peiyuan Zhu (Feb 10 2022 at 22:34):Ok maybe I'm missing the point. So the difficulty here seems to be what's referred to as "combinatorial underpinnings of linear algebra". So we're interested in a combinatorial perspective of linear algebra that's free of real numbers, am I correct?
John Baez (Feb 10 2022 at 22:35):Right: trying to do what people often do with rational, real or complex numbers using just combinatorics (stuff like finite sets, finite groupoids, etc.).
Peiyuan Zhu (Feb 10 2022 at 22:59):What is the benefit of doing so? Are we aiming for a systematic generalization of the things we used to do with rational/real/complex numbers?
John Baez (Feb 10 2022 at 23:12):At the time I was noticing that a lot of fancy applications of linear algebra to group representation theory really does have a combinatorial origin.
John Baez (Feb 10 2022 at 23:12):This is an interesting fact.
John Baez (Feb 10 2022 at 23:13):And separately, I was hoping to explain the role of real and complex numbers in physics in terms of combinatorics.
Peiyuan Zhu (Feb 11 2022 at 00:35):There is this paragraph from the page https://math.ucr.edu/home/baez/groupoidification/#episode3: "The reason is that incidence geometries usually have interesting symmetries, and a groupoid is like a "set with symmetries". For example, consider lines in 3-dimensional space. These form a set, but there are also symmetries of 3-dimensional space mapping one line to another. To take these into account we need a richer structure: a groupoid!". A group action acting on a set with a group is also a set with symmetries. Are we referring to the fact that group actions form a groupoid here to motivate the definition of a groupoid?
John Baez (Feb 11 2022 at 00:45):I'm not trying to motivate the definition of groupoid, which is very simple: it's a category where all morphisms are isomorphisms.
I'm trying to explain how 'sets with symmetries' are related to groupoids.
Here's the key fact:
If you give me the action of a group on a set I can build a groupoid called the 'action groupoid' or 'weak quotient' .
I explain this fact in https://math.ucr.edu/home/baez/groupoidification/#episode3, which you are in the process of reading!
John Baez (Feb 11 2022 at 00:45):When you understand what I wrote, you'll understand this motto:
A GROUPOID EQUIPPED WITH CERTAIN EXTRA STUFF IS THE SAME AS A GROUP ACTION
David Egolf (Feb 11 2022 at 01:38):Oh, cool! If I understand, the idea that is that the 'action groupoid' is sort of like a flow chart relating how an action of on the set changes different set elements in to each other. The objects of the groupoid are the different elements of , and you put an arrow (label it ) from one element to another if there is some that changes in to when it acts on . (So that the bijective map on corresponding to sends to ).
Peiyuan Zhu (Feb 11 2022 at 01:57):John Baez said:
I'm not trying to motivate the definition of groupoid, which is very simple: it's a category where all morphisms are isomorphisms.
I'm trying to explain how 'sets with symmetries' are related to groupoids.
Here's the key fact:
If you give me the action of a group on a set I can build a groupoid called the 'action groupoid' or 'weak quotient' .
I explain this fact in https://math.ucr.edu/home/baez/groupoidification/#episode3, which you are in the process of reading!
Is this understandable from a 'geometric morphism' perspective where the weak quotient being the slice categories and the geometric morphism being isomorphisms? The direction of the arrows seem to point to the opposite tho so more like an 'under category'.
Zhen Lin Low (Feb 11 2022 at 02:24):That notation does not indicate a slice category here...
Peiyuan Zhu (Feb 11 2022 at 02:26):David Egolf said:
Oh, cool! If I understand, the idea that is that the 'action groupoid' is sort of like a flow chart relating how an action of on the set changes different set elements in to each other. The objects of the groupoid are the different elements of , and you put an arrow (label it ) from one element to another if there is some that changes in to when it acts on . (So that the bijective map on corresponding to sends to ).
Yea I think it's pretty cool too. I think this can be understood as a 'relationalism view of symmetry'?
Peiyuan Zhu (Feb 11 2022 at 02:27):Zhen Lin Low said:
That notation does not indicate a slice category here...
Good to know. I just sense that they are talking about vaguely similar things but I can't tell whether there's something indeed. Maybe it's just the idea of relationalism.
Peiyuan Zhu (Feb 11 2022 at 02:55):This paragraph "Given this, what's the meaning of a span of groupoids? You could say it's a "invariant" witnessed relation - that is, a relation with witnesses that's preserved by the symmetries at hand. These are the very essence of incidence geometry. For example, if we have a point and a line lying on a plane, we can rotate the whole picture and get a new point and a new line lying on a new plane. Indeed, a "symmetry" in incidence geometry is precisely something that preserves all such "incidence relations"." Is it talking about transformations of the entire space while leaving the relative positions of the point, line, and plane intact? I think I'm a bit lost when this page talks about incidence geometry.
And I think the links to week 106, 145, and 178 have expired. image.png
David Egolf (Feb 11 2022 at 03:51):I think this link will work for week 106: https://math.ucr.edu/home/baez/week106.html @Peiyuan Zhu
John Baez (Feb 11 2022 at 05:27):All the stuff in
https://math.ucr.edu/home/baez/groupoidification/#episode3
is from This Week's Finds, but I collected it and moved it to the subdirectory /groupoidification, so the links don't work anymore. I'll fix the links.
Okay, they should be fixed now.
John Baez (Feb 11 2022 at 05:30):David Egolf said:
Oh, cool! If I understand, the idea that is that the 'action groupoid' is sort of like a flow chart relating how an action of on the set changes different set elements in to each other. The objects of the groupoid are the different elements of , and you put an arrow (label it ) from one element to another if there is some that changes in to when it acts on . (So that the bijective map on corresponding to sends to ).
Exactly. Yes, it's pretty clear if you draw a picture of it, or imagine the picture mentally.
John Baez (Feb 11 2022 at 05:31):Is this understandable from a 'geometric morphism' perspective where the weak quotient being the slice categories and the geometric morphism being isomorphisms?
That doesn't make sense to me; I'm not talking about slice categories or geometric morphisms here.
Peiyuan Zhu (Feb 13 2022 at 07:42):Is there any good reference on what does algebraic topology have to do with logic?
Peiyuan Zhu (Feb 13 2022 at 07:48):Is there any good survey paper of how group invariance is made more general via algebraic topology?
Leopold Schlicht (Feb 13 2022 at 13:56):Peiyuan Zhu said:
Is there any good reference on what does algebraic topology have to do with logic?
https://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf
Peiyuan Zhu (Feb 13 2022 at 15:30):Leopold Schlicht said:
Peiyuan Zhu said:
Is there any good reference on what does algebraic topology have to do with logic?
https://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf
Sure, I just remember some people told me it's just an analogy but not like Boole's work arguing algebra is more fundamental than logic, but maybe this was taken too literally.
How did topology enter into the picture of logic in history? For my entire life I never thought that they're related anyhow until encountering HoTT. Boolean algebra somehow is easier to see and well-known. Is there an "intuitive" reason for this?
Leopold Schlicht (Feb 13 2022 at 16:30):I think https://www.youtube.com/watch?v=m_PecfbEWik&ab_channel=Computerphile explains how groupoids occured in the study of Martin-Löf type theory (roughly speaking, groupoids constitute a model of Martin-Löf type theory). And then somebody had the idea to consider -groupoids instead of groupoids. And this suggested new axioms that one can add to Martin-Löf type theory, such as univalence or higher inductive types.
Leopold Schlicht (Feb 13 2022 at 16:32):Oh, and -groupoids are basically the same as (homotopy types of) spaces, by Grothendieck's homotopy hypothesis.
Leopold Schlicht (Feb 13 2022 at 16:35):That's why HoTT can be used as a foundation of mathematics in which one can do homotopy theory "synthetically".
Leopold Schlicht (Feb 13 2022 at 16:43):Another relationship between logic and topology: the category of Boolean algebras is dually equivalent to the category of Stone spaces (Stone duality).
Leopold Schlicht (Feb 13 2022 at 16:45):Yet another: Grothendieck topoi are "generalized topological spaces". There is a bijection between Grothendieck topoi up to equivalence and geometric theories up to Morita-equivalence.
Leopold Schlicht (Feb 13 2022 at 16:49):(Geometric theories belong to the world of logic.)
Leopold Schlicht (Feb 13 2022 at 17:09):Leopold Schlicht said:
Another relationship between logic and topology: the category of Boolean algebras is dually equivalent to the category of Stone spaces (Stone duality).
Since you are interested in history, check out the historical introduction in Johnstone's book on Stone spaces. I think there he says something about how Stone discovered his duality.
Peiyuan Zhu (Feb 13 2022 at 17:16):Thanks!!
Peiyuan Zhu (Feb 24 2022 at 00:32):Is there any situation where "foreset" and "afterset" structure being important in applied category theory? The definition is a follows, let be a relation between and . "foreset" is the elements of that's related via , that is , and "afterset" of is the elements of that is related by , that is, ?
John Baez (Feb 24 2022 at 00:49):People often study these when is a partial order. Then they're called the 'lower set' and 'upper set'.
John Baez (Feb 24 2022 at 00:52):They are incredibly important in the study of posets (partially ordered sets); for example there's a version of the Yoneda embedding where you embed any poset into its poset of upper sets.
I don't have anything interesting to say about their appearance in applied category theory. But Fong and Spivak's book Seven Sketches is a nice introduction to applied category theory that starts by focusing on posets.
Peiyuan Zhu (Feb 24 2022 at 02:08):I noticed Galois connection can pass on properties from one poset to another. For example, this can be formed by considering the closure dual interior operators of the lattice. I wonder if the same can be said for groups instead of lattices. For example, a "closure" operator on two groups that finds the least inclusive supergroup on some space that is weaker than a group locally.
In Modal HoTT book I noticed the sentence that by moving up the type hierarchy, "necessity" of a groupoid is invariant under group action (but I don't know what "possibility" is in this case?) I guess this is something relevant to what I'm looking for?
I don't know if my question makes sense. If not, please let me know.
John Baez (Feb 24 2022 at 02:23):Peiyuan Zhu said:
I noticed Galois connection can pass on properties from one poset to another. For example, this can be formed by considering the closure dual interior operators of the lattice. I wonder if the same can be said for groups instead of lattices. For example, a "closure" operator on two groups that finds the least inclusive supergroup on some space that is weaker than a group locally.
I don't know what you mean by that. There's a lattice of subgroups of a group, and this is fundamental to Galois theory. But I don't know what the "least inclusive supergroup on a space that is weaker than a group locally". Please define that.
Samuel Steakley (Apr 09 2022 at 16:02):I have a very beginner question: In the category defined by the ordinal number 2 (i.e. the category with two objects and one non-identity morphism), I believe the non-identity morphism is both monic and epic. Is that true?
If so, then (some) faithful functors out of this category would provide obvious examples of faithful functors that do not preserve monos/epis.
Fawzi Hreiki (Apr 09 2022 at 18:00):Yes. In any poset all maps are both monic and epic. This is because there are no non trivial parallel pairs to worry about making equal.
Jules Hedges (Apr 10 2022 at 11:26):I never knew that!
Sophie El Agami (Apr 19 2022 at 17:52):Hello there! I have some questions about čech cohomology of sheaves... Could anyone explain these to me? I hope I'm wording myself right :sweat: :thinking:
The question is inspired by the stack projects definition of an Algebraic structures (6.15.1). Let be any complete abelian category and be the category of abelian groups and be an exact faithful functor, that preserves limits and filtered colimits. If the functor has a left adjoint, then the cateogry is equivelent to the category of -modules for some ring. If the categort has a generator, then such left adjoint exists. Under waht condetions does there exist a left adjoint? Does one always exist?
Another queestion is about sheaves. If a topological space is hausdorff and paracompact, then the cech cohmology coincides with sheaf cohomology. Why is the space required to be hausdorff? I am following the proof in Bredon's sheaf theory and it appears to me that the only place where the paracompactness is used is in proposition 4.3. In this propositon, it is proved that every element in the čech cohomology is represented by an element in the čhech complex of an locally finite covering. So my question is, why is it required for the space to be hausdorff. Is it true in general, that the subcategory of locally finite open covering is a final subcategory of the category of open covering (seen as the poset category with the refinement relation) for a paracompact space ?
John Baez (Apr 19 2022 at 18:25):I don't know the answer to your second question, but it's very nice. To my naive eyes it seems like an elegant definition of paracompactness to say that locally finite open covers form a final subcategory of the poset of open covers (with the refinement relation as the partial ordering).
Jens Hemelaer (Apr 19 2022 at 22:26):For your second question, Grothendieck has given an example of a compact space (in particular paracompact) such that Čech cohomology is different from sheaf cohomology. It starts at the end of page 177 in his Tohoku paper (or page 61 of the English translation). I got the reference from MathOverflow here. I don't see at the moment where Hausdorffness is used in Bredon's proof.
Sophie El Agami (Apr 20 2022 at 19:04):John Baez said:
I don't know the answer to your second question, but it's very nice. To my naive eyes it seems like an elegant definition of paracompactness to say that locally finite open covers form a final subcategory of the poset of open covers (with the refinement relation as the partial ordering).
Well, that's a good thing I guess :upside_down:
Jens Hemelaer said:
For your second question, Grothendieck has given an example of a compact space (in particular paracompact) such that Čech cohomology is different from sheaf cohomology. It starts at the end of page 177 in his Tohoku paper (or page 61 of the English translation). I got the reference from MathOverflow here. I don't see at the moment where Hausdorffness is used in Bredon's proof.
Thanks for the reference - I found that Mathoverflow thread. I still don't understand why Hausdorffness is necessary though.
John Baez (Apr 20 2022 at 19:57):Neither do I. And I don't understand this counterexample yet.
It sounds like Bredon's proof (which I haven't looked at for years) hides the Hausdorffness assumption very carefully.
Morgan Rogers (he/him) (Apr 21 2022 at 06:34):Worth noting that paracompact includes Hausdorff in Bredon's book (see pg 21), so it's probably hidden in a much more basic fact about, say, the existence of a paracompactifying set of supports?
Matteo Capucci (he/him) (Apr 21 2022 at 07:23):Wikipedia says that Hausdorff + paracompact is the minimal setting to have partitions of unity, maybe that's important?
Patrick Nicodemus (Apr 21 2022 at 10:41):Matteo Capucci (he/him) said:
Wikipedia says that Hausdorff + paracompact is the minimal setting to have partitions of unity, maybe that's important?
Yes, it absolutely is.
Sophie El Agami (Apr 24 2022 at 19:05):Morgan Rogers (he/him) said:
Worth noting that paracompact includes Hausdorff in Bredon's book (see pg 21), so it's probably hidden in a much more basic fact about, say, the existence of a paracompactifying set of supports?
If it is about the paracompactifying set of supports, then I could not find it. However, I think it might have to do with finding a restricting such that , since if we had it follows without a problem from the stalks being zero. On the other hand, if , then the stalks alone can't show it and we need some condition on the space. In other words we need , which is not clear if , or at least not to me... What do you think?
Peiyuan Zhu (Apr 28 2022 at 22:19):Fabrizio Genovese said:
I agree with John, you are probably not ready. But just to give a simple example, you may say that
"A (small) category consists of a set of objects, a set of morphisms, and some laws and axioms that must hold."
In this case, you are considering the theory of sets as primitive, start from there and say what categories are. But there are also ways to axiomatize what categories are without resorting to set theory. Ad that point, you may say that
"A set is a (small) category where there are only identity arrows. A function between set is a functor between such categories."
I put the "small" thing in parentheses because it's not important to focus on it right now.
...Here you are using category theory as primitive and defining set theory using that. Who comes first? Well, it depends on your point of view! The point is exactly that many concepts in category theory can be defined in multiple ways, that can be proved equivalent. Once you pick one, you can use it to define other concepts, or vice versa, a bit like in the example above
Revisiting a previous discussion. What about a type in HoTT? What's the difference between a type and a set? (and a category?)
Peiyuan Zhu (Apr 30 2022 at 05:00):Mike Shulman said:
For the second question, we say that a statement or property is intenSional (not intenTional) if it depends on how things are defined, and extensional if it depends only on what those things are. For instance, the morning star and the evening star are intensionally different, but extensionally equal (both being the planet Venus). Mathematically, "" and "the number of exotic 7-spheres" are intensionally different, but extensionally equal.
You may have heard these words used to describe type-theoretic foundations of mathematics. In this case, for historical reasons, their use is extremely confusing and often almost "backwards" from what it ought to be. In type-theoretic foundations, there are (at least) two different kinds of "equality": a "judgmental" equality and a "typal" (or, more traditionally, "propositional") equality. The typal equality is the "real" equality of mathematics, which is usually extensional: "" and "the number of exotic 7-spheres" are typally equal. (It's also the equality that is expanded by the univalence axiom to mean "isomorphism".) To be sure, sometimes one considers type theories in which the typal equality is not fully determined, e.g. function extensionality may fail, so that it is not as extensional as it might be; but it is always at least somewhat extensional.
The judgmental equality is potentially distinct from the typal equality, and thus potentially intensional. One says that an "intensional type theory" is one in which the judgmental equality is truly intensional, whereas an "extensional type theory" is one in which the judgmental and typal equality coincide and are thus both extensional. Unfortunately, this sometimes leads people to apply these adjectives to the typal equality itself and speak of an "intensional equality type" or an "extensional equality type", whereas in fact as I said the typal equality is always extensional (or at least aspires to be so) and the adjective really refers to the behavior of the judgmental equality.
Ironically, the univalence axiom, which is an ultimate sort of extensionality property for the typal equality, is inconsistent with extensional type theory. Thus, it is intensional type theory that can actually be "more extensional", at least in the behavior of its typal equality.
Revisiting another old discussion: Can the extensionality vs intensionality distinction mentioned in this paper of Judea Pearl https://ftp.cs.ucla.edu/pub/stat_ser/r107-shrobe.pdf be understood in the same sense?
Peiyuan Zhu (May 04 2022 at 23:49):John Baez said:
Yeah, category theory is deeply connected to relationalism: the Yoneda lemma is sometimes seen as way to make the idea of relationalism precise.
Another old discussion. Do the words "relationalism" and "structural realism" often have interchangeable meanings? https://plato.stanford.edu/entries/structural-realism/
Samuel Steakley (May 08 2022 at 02:59):In working out a duality argument for Category Theory in Context chapter 2, I have developed a claim that seems to admit an elementary proof, but I find that I'm feeling a little suspicious of myself today. Is this true?
Claim: given any covariant functor , we can define a contravariant functor as follows:
Reid Barton (May 08 2022 at 03:07):It depends on what you mean. Usually when people say "a contravariant functor " they really just mean a contravariant functor from to , or in other words a functor from to . If that's what you mean then the claim is false. But if you really mean a contravariant functor from (the opposite category of ) to , then it is true.
Samuel Steakley (May 08 2022 at 03:18):Ah, thank you! I see I was confusing those two meanings.
Nathanael Arkor (May 08 2022 at 11:31):Reid Barton said:
Usually when people say "a contravariant functor " they really just mean a contravariant functor from to , or in other words a functor from to .
I disagree that this is "usually" the case. The only people who say this are people who have confused the meaning of a contravariant functor: saying this is the "usual" interpretation normalises incorrect usage. On the other hand, I think introducing both covariant and contravariant functors is needlessly confusing; in practice, people rarely talk about contravariant functors.
Reid Barton (May 08 2022 at 12:07):I'm pretty sure that what I said is nevertheless correct.
John Baez (May 08 2022 at 17:08):I wouldn't say "people rarely talk about contravariant functors". It's true that when I'm out shopping and doing my errands I don't hear people talking about contravariant functors. But I don't hear them talking about functors at all! If you read math books and papers, you'll see plenty of talk about contravariant functors.
Todd Trimble (May 08 2022 at 17:20):Referring to Galois connections, I find it convenient to say that a relation induces a contravariant adjunction between and , so that I'm not obliged to pick which of those two receives the .
Mike Shulman (May 08 2022 at 17:23):It's true that the original meaning of "contravariant functor " was the same as "covariant functor " and that thus a "contravariant functor " would be the same as a covariant functor . But because it's so much clearer and more convenient to work with just plain functors, some of which have an opposite category in their domain, than to have separate notions of "covariant" and "contravariant" functor, in my experience no one uses "contravariant functor" in this sense any more.
If we accept that this is no longer the meaning of "contravariant functor", then the way becomes open to use the adjective in a different way without calling it incorrect. For instance, we could say that a "contravariant functor" is a functor whose domain is an opposite category. This usage doesn't change the domain of the functor when it's regarded as "contravariant", so that saying "a functor " and "a contravariant functor " mean exactly the same thing: the second one just emphasizes an aspect of the functor that is already present in the first one.
Mike Shulman (May 08 2022 at 17:24):The potential for confusion means that I probably wouldn't say, or recommend saying, "contravariant functor " at all. But I might say something like "...a functor . Since this is contravariant, ...".
Nathanael Arkor (May 08 2022 at 19:11):I think there's good reason to keep the original meanings, and that is that there are some 2-categories where you do not have a duality involution, and yet you have notions of covariant and contravariant functors (e.g. when enriching in a nonsymmetric monoidal category).
Nathanael Arkor (May 08 2022 at 19:12):(Mike's paper Contravariance through enrichment is of course very relevant.)
Reid Barton (May 08 2022 at 19:19):I'm not really interested in linguistic prescriptivism. From a Bayesian perspective, it is unlikely someone would say "a contravariant functor " when they could simply say "a functor ".
Reid Barton (May 08 2022 at 19:25):So it seems much more likely that but we're going to write where , are morphisms of .
Nathanael Arkor (May 08 2022 at 19:26):If someone started using "group" to mean a monoid, I'm not sure it'd be fair to call disagreeing with that usage "linguistic prescriptivism" rather than something like "definitional confusion".
Nathanael Arkor (May 08 2022 at 19:27):But I don't disagree the current terminology is easily confused.
Reid Barton (May 08 2022 at 19:27):Well, that would be a rather strange thing to do and not very much like the situation under discussion.
Nathanael Arkor (May 08 2022 at 19:28):The only point I want to make is that the notion of contravariant functor is genuinely a useful one, and I'd prefer it didn't become essentially meaningless.
Nathanael Arkor (May 08 2022 at 19:29):I'm not disagreeing that choice of terminology is subjective, and you may very well disagree with my opinion.
Nathanael Arkor (May 08 2022 at 19:31):It just feels a little frustrating to be called a linguistic prescriptivist after giving an objective, mathematical reason for preferring to keep a precise meaning of contravariant functor.
Mike Shulman (May 08 2022 at 23:23):Nathanael Arkor said:
I think there's good reason to keep the original meanings, and that is that there are some 2-categories where you do not have a duality involution, and yet you have notions of covariant and contravariant functors (e.g. when enriching in a nonsymmetric monoidal category).
Are you sure about that? It's not clear to me that it's any easier to define "contravariant functor" without at least a braiding on your base of enrichment than it is to define opposite categories.
Nathanael Arkor (May 09 2022 at 11:57):You're right: I had been thinking of the formalism in Linton's paper, but he only defines contravariant functors into the base of enrichment, not in general. In that case, off the top of my head I don't have an example of a notion of category that admits all contravariant functors but not opposites. I think the distinction is still useful formally, but that's likely not a sufficient justification for most.
Mike Shulman (May 09 2022 at 14:38):Note also that in the context of my "contravariance" paper that you mentioned, opposites are an absolute colimit. Thus, any 2-category with contravariance can be completed to a 2-category with opposites, in an idempotent way, analogously to the completion of an ordinary category under idempotent-splitting.
Nathanael Arkor (May 09 2022 at 14:54):Yes, though if you are interested in a concrete example of a 2-category with contravariance, it may not be the case that you can give a natural concrete description of the completion of the 2-category under these absolute colimits (rather than talking about "formal opposites"), which may be slightly dissatisfying. I agree that this construction makes assuming the existence of opposites relatively harmless, though.
Mike Shulman (May 09 2022 at 17:01):(To be sure, in another thread a few days ago, I was arguing that Cauchy completion isn't always innocuous...)
Naso (May 16 2022 at 01:57):I know it is false that 'a presheaf is a sheaf if and only if it preserves limits'. But is there at least a one-way implication there?
Morgan Rogers (he/him) (May 16 2022 at 07:22):The sheaf condition depends on a Grothendieck topology, an extra piece of data consisting of a collection of "covering sieves". Which presheaves are sheaves depends on the choice of Grothendieck topology.
For example, with the trivial Grothendieck topology (where only the maximal sieves are covering) every presheaf is a sheaf. At the other extreme, with the degenerate Grothendieck topology (where every sieve is covering), the only sheaf is the terminal presheaf, where all objects are mapped to the singleton set. Considering the walking arrow category, the presheaf which sends the initial object to and the terminal object to preserves arbitrary limits but is not a sheaf for the degenerate topology.
The category over which you're taking presheaves need not have given colimits, but even if you replace preservation of limits with a flatness condition, neither implication holds in general.
All of that said, you might be asking about the special case of sheaves on a locale/space, where the category involved is the category of opens of a space (which is cocomplete) and the Grothendieck topology is the canonical topology where the covering sieves are exactly the effective epimorphic ones. In this situation, the sheaf conditions effectively translate to demanding that "sieve-indexed" limits are preserved. For example, products aren't preserved: if I take with non-trivial intersection, will be not , but the pushout .
Preservation of limits is sufficient in this situation, but it's much stronger than the sheaf condition, so you won't find many sheaves this way.
Naso (May 16 2022 at 08:28):wonderful answer, thanks a lot Morgan
Naso (May 18 2022 at 09:23):Does this define a functor from simplicial sets to sets? where , and where is the unique nondegenerate simplex of such that there exists a surjection with (this comes from the 'Eilenberg-Zilber lemma').
Reid Barton (May 19 2022 at 17:29):It seems to me that it does, which I find a bit unintuitive somehow.
Naso (May 19 2022 at 23:49):Thanks for checking, Reid!
Naso (May 21 2022 at 00:47):I have two questions about Riehl's article 'A LEISURELY INTRODUCTION TO SIMPLICIAL SETS' (https://math.jhu.edu/~eriehl/ssets.pdf).
Section 4 is titled 'A unified presentation of all adjunctions' (whose left adjoint has domain sSet). There is a general construction where given a functor , another functor is defined by . Then it is shown that has a left adjoint given by left kan extension of along the Yoneda embedding.
Why is every adjunction whose left adjoint has domain given by this formula?
Why do all the useful adjunctions with seeem to have the domain of the left adjoint and not the right? I guessed it has something to do with the density theorem, and left adjoints preserving colimits..?
Morgan Rogers (he/him) (May 21 2022 at 07:56):The category of simplicial sets is the "free colimit completion of ", meaning that every simplicial set can be presented as a colimit of simplices in an unconstrained way. Making the word 'unconstrained' precise is a little tricky, but it amounts to saying that any functor from to a cocomplete category extends uniquely (up to isomorphism) to a cocontinuous functor with domain . That category satisfies all of the conditions of the special adjoint functor theorem, and the definition of is a special case of the construction of an adjoint to a cocontinuous functor from that theorem. Note that it (only) requires to be locally small to make sense.
John Baez (May 21 2022 at 22:44):It may be helpful to note that nothing in Naso's question, or Morgan's answer, is special to . We could replace by any category and replace by , and similar results would hold.
John Baez (May 21 2022 at 22:45):There are a lot of cool things special to , but they're not showing up here.
Naso (May 23 2022 at 01:52):Thank you Morgan & John, this answers my first question. I'm still wondering a bit about Q2. Is there a simple heuristic to understand why adjunctions with domain of the right adjoint are not interesting (if that is really true)?
As a random example, precomposition with the inclusion of injective monotone functions into has a left adjoint given by left Kan extension described here (https://kerodon.net/tag/00XT). Right Kan extension also gives a right adjoint to precomposition with , but it is not mentioned there or anywhere else I looked.
Todd Trimble (May 23 2022 at 02:33):Well, it's far from true that right adjoints out of are uninteresting. I'll direction your attention here, to discussions of simplicial skeleta and coskeleta.
Morgan Rogers (he/him) (May 23 2022 at 08:37):Specializing to a topos theory perspective, sSet "classifies" the theory of total orders with a (distinct) tip and bottom element. So an adjunction between sSet and another topos E, where the left adjoint preserves finite limits and has domain sSet is a geometric morphism, and corresponds to a model of this theory (that is, a bounded total order) in E, and these are relatively easy to find.
All results about classifying toposes go this way around: to easily construct a geometric morphisms out of sSet into E, you need to identify a theory classified by E. Since that can be arbitrarily complicated, it's not as easy to find those.
Naso (May 25 2022 at 00:22):Thanks for the great 'counterexample', Todd :)
Morgan thanks, I like this explanation. Is it related somehow to the fact that is the 'universal category-equipped-with-a-monoid'?
I'm not sure what you mean about 'total orders with distinct top and bottom' since has total orders of length one. Can you say something more about that please?
Morgan Rogers (he/him) (May 25 2022 at 05:58):It's someday related to the fact about , although that correspondence is for monoidal functors.
The 1-simplex , viewed as an object of , carries the structure required to make it a total order: the two morphisms form a relation which one can check is reflexive, total and antisymmetric, and it is transitive thanks to all of the higher simplices. The two morphisms are distinct constants which are necessarily the top and bottom elements of the total order. All of this structure is preserved under finite limits, so the image of becomes an object equipped with the structure of a total order with top and bottom elements (which are distinct as soon as the initial object is also preserved).
Peiyuan Zhu (Jun 01 2022 at 20:00):In the paper The Rosetta Stone, John says "Computation entered the game around 2000, when Freedman, Kitaev, Larsen and Wang [41] showed that certain systems of anyons could function as ‘universal quantum computers’". I wonder how this project has been going and whether there will be ways to reach out to these people? Is it correct to understand that this approach is completely different from what D-Wave is doing?
Martti Karvonen (Jun 01 2022 at 20:29):The keyword is topological quantum computing, and a selling point for it is noise resistance. AFAIK, it hasn't been progressing experimentally as fast as the other approaches, but I do hear claims about experimental progress every now and then so maybe it'll pan out eventually?
John Baez (Jun 02 2022 at 20:40):Yes, topological quantum computing as studied by Freedman-Kitaev-Larsen-Wang and Microsoft's Station Q is completely different from what D-Wave is doing.
John Baez (Jun 02 2022 at 20:44):As you'll see from the link, D-Wave has been doing "quantum annealing", not trying to build a universal quantum computer.
(Starting last year they started trying to build one, but they are famous for making their work sound impressive, so don't take this seriously until they actually accomplish something!)
Peiyuan Zhu (Jun 02 2022 at 21:53):I'm thinking of an extension of information geometry / evolutionary game as posted here https://math.ucr.edu/home/baez/information/. Normally in information geometry we work on a manifold of parametric family of distributions . I wonder what will happen if we take the auxiliary variable used for random number generation into account. As an example, we may generate uniform random variable and compare it with in order to obtain a Bernoulli random variable . is mapped from a deterministic function that takes value if and if . So here we consider the manifold of structural equations each defining a probability distribution on sets by "inversion" , and statistical inference comes down to [1] a way of combining these structural equations / distribution on sets and output a new distribution on sets and [2] computing the probability of these random sets intersecting a given set as an upper estimate and the probability of a given set is contained in the random sets as a lower estimate. What are the questions that can be interesting to ask in this case?
In fact, I would be interested in hearing how generally people approach mathematical research. I did a statistics thesis but I felt like I was heavily reliant on my supervisor for intuition at that time, and get the sense of being dragged along throughout publishing that paper. I had a math degree but I felt like it was mostly me trying to pass the exams efficiently. What are the question that mathematicians ask themselves when they read a mathematical paper? I assume it isn't just about whether so and so is true, but instead we can ask what's the structure of these mathematical objects and what we can do about them?
Todd Trimble (Jun 02 2022 at 22:31):How people approach research is partly a matter of how long they've been doing research, but there are things people can do just to get engaged and eventually sucked in (fascinated), which is a precondition for doing significant research, and many are things anyone can do at any stage of mathematical life. One is to seek out the really telling examples. Another is logical/analytical: if you have a theorem and its proof that you want to understand better, play little games like: could the hypotheses be weakened? Where do we use this hypothesis? What if I changed the hypotheses a little? Could the conclusion be strengthened? Is the converse true?
In my own case, a lot of thinking is stimulated by a sense that something could be phrased better, or at least phrased in a way I find more satisfying. Even such a little thing as developing notation that I personally prefer helps me. Anything that creates a feeling of doing something, even if it's a little thing, helps. In short, making mathematics your own, even if it's in tiny ways, is vital.
After you gain more experience, you can set yourself little challenges. You see a result stated. Not necessarily a big result, maybe a little lemma. But then don't look at the proof -- ask yourself: how might I prove this? Train yourself on little problems that naturally come your way, and develop a little independence. Don't google every damned thing, at least not before giving it the old college try yourself.
If you're reading a proof of something you really want to know, go over and over it. Once you think you've got it clear and settled in your mind, write up a note about it, preferably without peeking back at the proof.
Collect a bank of pictures and examples and lemmas. Oh, and counterexamples. Create over time your own personalized conceptual toolbox.
Write for other people, or as if for other people. With the idea "if I can understand this, then I can help others understand this".
If you have to break away from mathematics, try to end on a positive note. Some people (if they're writing) even stop mid-sentence, so that they can pick up right where they left off.
This is all off the top of my head, and it's stuff most anyone can put into practice, no matter what stage of development they are.
Todd Trimble (Jun 02 2022 at 22:43):A few more: make up your own problems. Be willing to goof around. Don't ever think you're conceptually too sophisticated to play around with numbers or finite, small structures.
Peiyuan Zhu (Jun 02 2022 at 22:56):Peiyuan Zhu said:
I'm thinking of an extension of information geometry / evolutionary game as posted here https://math.ucr.edu/home/baez/information/. Normally in information geometry we work on a manifold of parametric family of distributions . I wonder what will happen if we take the auxiliary variable used for random number generation into account. As an example, we may generate uniform random variable and compare it with in order to obtain a Bernoulli random variable . is mapped from a deterministic function that takes value if and if . So here we consider the manifold of structural equations each defining a probability distribution on sets by "inversion" , and statistical inference comes down to [1] a way of combining these structural equations / distribution on sets and output a new distribution on sets and [2] computing the probability of these random sets intersecting a given set as an upper estimate and the probability of a given set is contained in the random sets as a lower estimate. What are the questions that can be interesting to ask in this case?
In fact, I would be interested in hearing how generally people approach mathematical research. I did a statistics thesis but I felt like I was heavily reliant on my supervisor for intuition at that time, and get the sense of being dragged along throughout publishing that paper. I had a math degree but I felt like it was mostly me trying to pass the exams efficiently. What are the question that mathematicians ask themselves when they read a mathematical paper? I assume it isn't just about whether so and so is true, but instead we can ask what's the structure of these mathematical objects and what we can do about them?
Essentially it is a game like this. Define a relation between the known and the unknown and the random seed via an equation. Repeat independent experiments to collect known, in order to estimate the sets of the unknowns. Your data is the result of a certain relation between the random seed and the unknown parameter instead of "following some distribution". Is there anything in geometry around the lines of "relational geometry" or "constructive geometry"? Does this look like something that has to do with "non-absolute space" in physics? You don't start with a prescribed manifold, you construct a space from relations.
Peiyuan Zhu (Jun 02 2022 at 22:58):Todd Trimble said:
A few more: make up your own problems. Be willing to goof around. Don't ever think you're conceptually too sophisticated to play around with numbers or finite, small structures.
Right. I always find myself severely underestimating the intricacy of small objects, thereby skipping important details and intuitions. What do you think about applications? Do you think about things happening in your everyday social life to be fruit for mathematically musing? Or is this helpful?
Peiyuan Zhu (Jun 02 2022 at 23:13):Peiyuan Zhu said:
I'm thinking of an extension of information geometry / evolutionary game as posted here https://math.ucr.edu/home/baez/information/. Normally in information geometry we work on a manifold of parametric family of distributions . I wonder what will happen if we take the auxiliary variable used for random number generation into account. As an example, we may generate uniform random variable and compare it with in order to obtain a Bernoulli random variable . is mapped from a deterministic function that takes value if and if . So here we consider the manifold of structural equations each defining a probability distribution on sets by "inversion" , and statistical inference comes down to [1] a way of combining these structural equations / distribution on sets and output a new distribution on sets and [2] computing the probability of these random sets intersecting a given set as an upper estimate and the probability of a given set is contained in the random sets as a lower estimate. What are the questions that can be interesting to ask in this case?
In fact, I would be interested in hearing how generally people approach mathematical research. I did a statistics thesis but I felt like I was heavily reliant on my supervisor for intuition at that time, and get the sense of being dragged along throughout publishing that paper. I had a math degree but I felt like it was mostly me trying to pass the exams efficiently. What are the question that mathematicians ask themselves when they read a mathematical paper? I assume it isn't just about whether so and so is true, but instead we can ask what's the structure of these mathematical objects and what we can do about them?
In other words, how can a Riemanian manifold be constructed via causal / near causal relations.
Naso (Jun 03 2022 at 00:34):under good conditions, precomposing with a functor defines a functor that has left/right adjoints described by Kan extensions. Postcomposing with defines a functor . is there also a theory that explains when this has adjoints?
Naso (Jun 03 2022 at 01:09):naso evangelou-oost said:
under good conditions, precomposing with a functor defines a functor that has left/right adjoints described by Kan extensions. Postcomposing with defines a functor . is there also a theory that explains when this has adjoints?
apparently the keyword is 'Kan lift' https://ncatlab.org/nlab/show/Kan+lift
John Baez (Jun 03 2022 at 02:35):What are the question that mathematicians ask themselves when they read a mathematical paper?
First I want to make sure I understand the definitions of all the key terms, i.e. the words that appear in the main theorems. By "understand them" I mean:
1) I know them by heart - if someone wakes me up in the middle of the night I can say these definitions.
2) I can explain why the definitions are interesting.
3) I can give examples of things obeying those definitions and things that almost obey them but not quite.
Second I try to understand what the main theorems say. This means:
1) I know examples of things obeying the hypotheses of the theorem (and thus the conclusions).
2) I know examples of things that obey all but one of the hypotheses, and not the conclusions.
3) I can explain why the theorem is interesting. This includes being able to say why the theorem is believable... or, alternatively, why it's shocking. It also includes having some idea of what the theorem is used for.
John Baez (Jun 03 2022 at 02:37):Then, last, I go through the proofs of the theorems. I only do this in detail if I am seriously interested in the subject. The goal is to understand the proof so that I can reinvent the proof, not just memorize the proof. But I may settle for getting a rough idea of the proof.
Todd Trimble (Jun 03 2022 at 02:44):Peiyuan Zhu said:
Todd Trimble said:
A few more: make up your own problems. Be willing to goof around. Don't ever think you're conceptually too sophisticated to play around with numbers or finite, small structures.
Right. I always find myself severely underestimating the intricacy of small objects, thereby skipping important details and intuitions. What do you think about applications? Do you think about things happening in your everyday social life to be fruit for mathematically musing? Or is this helpful?
I tend not to think along such lines; perhaps that makes me an outlier on this channel. I sort of have an eye out for potential applications where I might be able to put my fairly specialized skills to work, but I'm happy to let that happen organically and not force it. I am impressed by some people here who routinely make that happen. It takes a certain breadth of scientific outlook to pull it off.
I allow myself to think about things off the beaten track for me where I suspect category theory could possibly give me an inside edge. Recently I'd been thinking about things connected with matchings in graph theory (Hall's marriage, Birkhoff-von Neumann, Ford-Fulkerson, etc.) which feel like they have compositional flavor. I'm happy just to let that murky intuition percolate for a while.
I can't figure out what your question about Riemannian manifolds is driving at. I do know what a Riemannian manifold is.
Peiyuan Zhu (Jun 03 2022 at 04:04):Was wondering if @John Baez has something to comment on the information geometry part
Peiyuan Zhu (Jun 03 2022 at 04:06):Todd Trimble said:
Peiyuan Zhu said:
Todd Trimble said:
A few more: make up your own problems. Be willing to goof around. Don't ever think you're conceptually too sophisticated to play around with numbers or finite, small structures.
Right. I always find myself severely underestimating the intricacy of small objects, thereby skipping important details and intuitions. What do you think about applications? Do you think about things happening in your everyday social life to be fruit for mathematically musing? Or is this helpful?
I tend not to think along such lines; perhaps that makes me an outlier on this channel. I sort of have an eye out for potential applications where I might be able to put my fairly specialized skills to work, but I'm happy to let that happen organically and not force it. I am impressed by some people here who routinely make that happen. It takes a certain breadth of scientific outlook to pull it off.
I allow myself to think about things off the beaten track for me where I suspect category theory could possibly give me an inside edge. Recently I'd been thinking about things connected with matchings in graph theory (Hall's marriage, Birkhoff-von Neumann, Ford-Fulkerson, etc.) which feel like they have compositional flavor. I'm happy just to let that murky intuition percolate for a while.
I can't figure out what your question about Riemannian manifolds is driving at. I do know what a Riemannian manifold is.
Riemannian manifold is a manifold first (which I assume you know). Then it has a Riemannian metric that's positive definite, operating on the tangent spaces.
Todd Trimble (Jun 03 2022 at 11:26):I said I do know what a Riemannian manifold is.
John Baez (Jun 03 2022 at 11:57):Peiyuan Zhu said:
Was wondering if John Baez has something to comment on the information geometry part
I don't have much to say about information geometry beyond the 20 blog articles that I wrote about it. Most of what I'd say is contained there... and it sounds like you've read those, or some of those.
John Baez (Jun 03 2022 at 12:04):I was trying to learn information geometry when I wrote those articles. As you can see, I was doing what @Todd Trimble recommends: playing around with ideas, trying to make them my own by putting a new spin on them. Instead of focusing on parametric statistics as people often do, I was focusing on the manifold of all probability measures, which is the "universal" form of parametric statistics (in the categorical sense). I was thinking about the geometry of this manifold, mainly made into an infinite-dimensional manifold via the Fisher metric. I was trying to understand the Fisher metric in terms of concepts I already understood, like relative entropy. And I was trying to understand what this has to do with entropy and free energy in physics. Later, I was trying to understand Marc Harper's work on information geometry and evolutionary game theory.
John Baez (Jun 03 2022 at 12:05):There's a lot of basic stuff about information geometry that I don't really understand. Maybe sometime I'll go back and try to think about it.
Zhen Lin Low (Jun 03 2022 at 14:09):I tried to convince myself that the Fisher metric is the "only thing it can be" but I didn't succeed. Is there a way to deduce the metric from the universal property?
John Baez (Jun 03 2022 at 14:19):No.
John Baez (Jun 03 2022 at 14:21):The Fisher metric on probability distributions on an n-element set is the one that makes the space of probability measures isometric to the intersection of the unit sphere in and the orthant .
John Baez (Jun 03 2022 at 14:23):But there are other Riemannian metrics on this manifold with corners with the same symmetry group.
Todd Trimble (Jun 03 2022 at 14:46):Peiyuan Zhu said:
Todd Trimble said:
A few more: make up your own problems. Be willing to goof around. Don't ever think you're conceptually too sophisticated to play around with numbers or finite, small structures.
Right. I always find myself severely underestimating the intricacy of small objects, thereby skipping important details and intuitions. What do you think about applications? Do you think about things happening in your everyday social life to be fruit for mathematically musing? Or is this helpful?
Peiyuan Zhu said:
In other words, how can a Riemanian manifold be constructed via causal / near causal relations.
I really don't know how the "in other words" is supposed to be segueing from "things happening in your everyday social life" to Riemannian manifolds that are somehow constructed via causal / near causal relations.
But since you asked how people do research, which includes two people trying to talk to one another in a context meant to be conducive to doing research, it may be that @John Baez's advice about learning definitions would be more useful here. What I mean is that when it comes to conversations that include relatively sophisticated concepts like that of Riemannian manifold (normally not encountered until graduate school), both participants should really try their best to speak precisely, otherwise the conversation may not go far, or become frustrating for one or both parties. Some sort of mutual understanding of how terms are defined quickly becomes vital here.
So, I'm sorry, but I don't have much idea of what you're talking about, and I'm not that keen on trying to guess. Things in my everyday social life do not form a continuum of points. "Causal relationship" to me usually implies an asymmetry, and puts me more in mind of Lorentzian manifolds than Riemannian manifolds. Do you want to try again? (Of course, I may not be the best person to talk with here, but I'm okay with walking a few more steps.)
Zhen Lin Low (Jun 03 2022 at 15:07):But there are other Riemannian metrics on this manifold with corners with the same symmetry group.
Hmmm. That's disappointing. Or maybe I asked the wrong question...
Peiyuan Zhu (Jun 03 2022 at 15:55):Todd Trimble said:
Peiyuan Zhu said:
Todd Trimble said:
A few more: make up your own problems. Be willing to goof around. Don't ever think you're conceptually too sophisticated to play around with numbers or finite, small structures.
Right. I always find myself severely underestimating the intricacy of small objects, thereby skipping important details and intuitions. What do you think about applications? Do you think about things happening in your everyday social life to be fruit for mathematically musing? Or is this helpful?
Peiyuan Zhu said:
In other words, how can a Riemanian manifold be constructed via causal / near causal relations.
I really don't know how the "in other words" is supposed to be segueing from "things happening in your everyday social life" to Riemannian manifolds that are somehow constructed via causal / near causal relations.
But since you asked how people do research, which includes two people trying to talk to one another in a context meant to be conducive to doing research, it may be that John Baez's advice about learning definitions would be more useful here. What I mean is that when it comes to conversations that include relatively sophisticated concepts like that of Riemannian manifold (normally not encountered until graduate school), both participants should really try their best to speak precisely, otherwise the conversation may not go far, or become frustrating for one or both parties. Some sort of mutual understanding of how terms are defined quickly becomes vital here.
So, I'm sorry, but I don't have much idea of what you're talking about, and I'm not that keen on trying to guess. Things in my everyday social life do not form a continuum of points. "Causal relationship" to me usually implies an asymmetry, and puts me more in mind of Lorentzian manifolds than Riemannian manifolds. Do you want to try again? (Of course, I may not be the best person to talk with here, but I'm okay with walking a few more steps.)
Oh my bad the "causal relation" was meant to reply to my previous post. Just edited it. I'll try again.
Peiyuan Zhu (Jun 03 2022 at 15:57):Peiyuan Zhu said:
Peiyuan Zhu said:
I'm thinking of an extension of information geometry / evolutionary game as posted here https://math.ucr.edu/home/baez/information/. Normally in information geometry we work on a manifold of parametric family of distributions . I wonder what will happen if we take the auxiliary variable used for random number generation into account. As an example, we may generate uniform random variable and compare it with in order to obtain a Bernoulli random variable . is mapped from a deterministic function that takes value if and if . So here we consider the manifold of structural equations each defining a probability distribution on sets by "inversion" , and statistical inference comes down to [1] a way of combining these structural equations / distribution on sets and output a new distribution on sets and [2] computing the probability of these random sets intersecting a given set as an upper estimate and the probability of a given set is contained in the random sets as a lower estimate. What are the questions that can be interesting to ask in this case?
In fact, I would be interested in hearing how generally people approach mathematical research. I did a statistics thesis but I felt like I was heavily reliant on my supervisor for intuition at that time, and get the sense of being dragged along throughout publishing that paper. I had a math degree but I felt like it was mostly me trying to pass the exams efficiently. What are the question that mathematicians ask themselves when they read a mathematical paper? I assume it isn't just about whether so and so is true, but instead we can ask what's the structure of these mathematical objects and what we can do about them?
In other words, how can a Riemanian manifold be constructed via causal / near causal relations.
Every important information that I wanted to say is contained in here. Is there anything that I'm missing?
Peiyuan Zhu (Jun 03 2022 at 15:58):So information geometry normally considers statistical manifold . We want a generalization of this.
Peiyuan Zhu (Jun 03 2022 at 16:00):Instead, we want to consider where is a deterministic function.
Peiyuan Zhu (Jun 03 2022 at 16:14):As an example: in information geometry a Bernoulli random variable is considered as where but here we want to consider where function
John Baez (Jun 03 2022 at 16:14):Peiyuan Zhu said:
So information geometry normally considers statistical manifold . We want a generalization of this.
Who is "we" and why do you want it?
John Baez (Jun 03 2022 at 16:15):By the way, the statistical manifold also requires , and this is important for the geometry.
Peiyuan Zhu (Jun 03 2022 at 16:16):Who is "we": Just me for now (and anyone who's interested in the problem) but I guess everyone else from imprecise probabilities / Desmpter-Shafer theory should want it too
Peiyuan Zhu (Jun 03 2022 at 16:16):John Baez said:
By the way, the statistical manifold also requires , and this is important for the geometry.
Right I'll fix it
Peiyuan Zhu (Jun 03 2022 at 16:24):Why do I want to do it: It's an important generalization called imprecise probability / Desmpter-Shafer theory. It's important because it's fully relational: the data is a function of a relation between parameter and "uniform randomness", i.e. a model is a relation between data, parameter, and "uniform randomness".
Peiyuan Zhu (Jun 03 2022 at 16:32):Ok having wrote this down I think I need to figure out if there're extra constraints on the function . I'll think about it a little bit more.
Peiyuan Zhu (Jun 03 2022 at 16:45):So far I can't think of anything except follows a distribution. I'll see how it constraints .
Peiyuan Zhu (Jun 03 2022 at 16:51):can be a CDF like the example above but it doesn't have to be. For example, multinomial random variable can be generated by Gumbel-Max trick other than inverse-CDF.
Peiyuan Zhu (Jun 03 2022 at 18:48):@Matteo Capucci (he/him) in a personal conversation mentioned this being the case for considering random variables directly but not deriving them from distributions. I remember Oliva Caramello told me to look at Alex Simpson's work. I think it start to make sense.
Peiyuan Zhu (Jun 04 2022 at 05:08):Hmm maybe not
Peiyuan Zhu (Jun 04 2022 at 05:10):I think I should start with Yoneda lemma and pinpoint the relationalism flavour of the setup first.
Peiyuan Zhu (Jun 08 2022 at 15:00):John Baez said:
Yes, topological quantum computing as studied by Freedman-Kitaev-Larsen-Wang and Microsoft's Station Q is completely different from what D-Wave is doing.
Are people able to simulate the thermodynamics of these topological quantum fields with classical computers? What kind of scientific computing techniques are people using in quantum physics / gravity nowadays?
John Baez (Jun 08 2022 at 15:32):I don't think people try to simulate the "thermodynamics" of topological quantum fields. Quantum computers are nowhere near thermal equilibrium.
John Baez (Jun 08 2022 at 15:33):What kind of scientific computing techniques are people using in quantum physics / gravity nowadays?
Quantum gravity is a tiny subject compared to "quantum physics" - the latter includes simulating individual molecules, which is a big topic in chemistry. People use lots of computing techniques in the quantum physics of chemistry.
John Baez (Jun 08 2022 at 15:35):Here is a list of computational chemistry software packages and here is an article on computational chemistry, a lot of which is really quantum physics.
Peiyuan Zhu (Jun 08 2022 at 23:10):Todd Trimble said:
Peiyuan Zhu said:
Todd Trimble said:
A few more: make up your own problems. Be willing to goof around. Don't ever think you're conceptually too sophisticated to play around with numbers or finite, small structures.
Right. I always find myself severely underestimating the intricacy of small objects, thereby skipping important details and intuitions. What do you think about applications? Do you think about things happening in your everyday social life to be fruit for mathematically musing? Or is this helpful?
Peiyuan Zhu said:
In other words, how can a Riemanian manifold be constructed via causal / near causal relations.
I really don't know how the "in other words" is supposed to be segueing from "things happening in your everyday social life" to Riemannian manifolds that are somehow constructed via causal / near causal relations.
But since you asked how people do research, which includes two people trying to talk to one another in a context meant to be conducive to doing research, it may be that John Baez's advice about learning definitions would be more useful here. What I mean is that when it comes to conversations that include relatively sophisticated concepts like that of Riemannian manifold (normally not encountered until graduate school), both participants should really try their best to speak precisely, otherwise the conversation may not go far, or become frustrating for one or both parties. Some sort of mutual understanding of how terms are defined quickly becomes vital here.
So, I'm sorry, but I don't have much idea of what you're talking about, and I'm not that keen on trying to guess. Things in my everyday social life do not form a continuum of points. "Causal relationship" to me usually implies an asymmetry, and puts me more in mind of Lorentzian manifolds than Riemannian manifolds. Do you want to try again? (Of course, I may not be the best person to talk with here, but I'm okay with walking a few more steps.)
I'll give you an example about what I mean by "social life". Yesterday I was at a beginner dance class. After class I was like "Hey did we just danced like a pair of adjoint monads".
Peiyuan Zhu (Jun 08 2022 at 23:23):Once upon a time I found body / action analogies very helpful in conceptualizing mathematics. For example, "pulling" a constant outside of an integral, "squeeze" with epsilon-delta, etc.
Naso (Jun 08 2022 at 23:32):What does a cosimplicial set "look like"? Since is equivalent to the category of finite intervals with distinct top and bottom, I guess it should be something made up of these intervals glued together? (https://ncatlab.org/nlab/show/simplex+category#DualityWithIntervals)
Patrick Nicodemus (Jun 08 2022 at 23:54):should be the free monoidal category generated by a distinguished comonoid, so for any comonoid there should be a way interpret a cosimplicial set as a combinatorial schema to glue this comonoid and its higher tensor powers together into something by taking a weighted colimit. Alternatively, given a monoid, it's a way to this monoid and its higher tensor powers together using a weighted limit. Maybe some insight will come from finding a good monoid or comonoid and computing some examples of the 'realization' of a cosimplicial set with respect to this monoid.
John Baez (Jun 08 2022 at 23:57):Patrick Nicodemus said:
should be the free monoidal category generated by a distinguished comonoid...
In math I say "should be" when I'm scared to say "is" - that is, when I'm not sure something is exactly true. So I'm happy to report that is the free strict monoidal category on a comonoid: given any strict monoidal category containing a comonoid , there's a unique strict monoidal functor sending and its comultiplication and counit to and its comultiplication and counit.
Naso (Jun 09 2022 at 00:09):Ok, but does this tell me what they should look like "visually"? Since cosimplicial sets are the free cocompletion of 'finite intervals with distinct top and bottom with morphisms as interval maps (monotone maps preserving top and bottom)', does this mean a cosimplicial set can be depicted by a bunch of these intervals glued together?
Mike Shulman (Jun 09 2022 at 00:10):To the extent that I try to visualize a cosimplicial set at all (which isn't much), I usually think of it more like a collection of "cosimplices" that are "coglued together".
Peiyuan Zhu (Jun 09 2022 at 01:21):What is the intuition of "natural transformations"? If functors are making analogies between views of processes, what are "natural transformations"? It looks to me commutativity implies that meanings are preserved / compatible between two analogies. But what makes natural transformations special? What is not a natural transformation?
Peiyuan Zhu (Jun 09 2022 at 01:21):B3ECAB0E-1D98-4F90-AFFD-BB2BE408754C.jpg
Peiyuan Zhu (Jun 09 2022 at 01:22):When I draw a diagram it seems like the two maps in blue always commute.
Peiyuan Zhu (Jun 09 2022 at 01:48):Also can anyone explain a bit how invariances are encoded into functors? Is it a notion that generalizes "group invariance"?
Matteo Capucci (he/him) (Jun 09 2022 at 11:29):Peiyuan Zhu said:
What is the intuition of "natural transformations"? If functors are making analogies between views of processes, what are "natural transformations"? It looks to me commutativity implies that meanings are preserved / compatible between two analogies. But what makes natural transformations special? What is not a natural transformation?
My go-to example of a non-natural transformation is the one you obtain by choosing an isomorphism for every finite-dimensional vector space : since this choice is non-canonical, this does not give you a natural transformation from the identity functor to the 'dual space' functor .
(This part is quite naive and speculative, if someone wants to correct my statements please do so)
It is an interesting fact that this example relies on this 'choice' process, which is underdetermined in a sense. It seems that every family of morphisms which is defined without making explicit reference to the 'internal structure' of (computer scientists say 'polymorphic on ' or 'parametric') doesn't give you problems when proving naturality. That's why it's called 'natural': it comes 'naturally' to us (cringe noises).
Peiyuan Zhu (Jun 09 2022 at 19:22):What are the pre-requisite to understand cohesive topos? Any physics that I need to know? Suppose I already know about categories, functors, natural transformations, Yoneda lemma, limits, colimits, adjunctions, monads, Kan extensions.
Peiyuan Zhu (Jun 09 2022 at 19:32):I think it might have something to do with physics because Lawvere talks about intensive vs extensive quantities. I'm not sure if has anything to do with those in thermodynamics https://en.wikipedia.org/wiki/Intensive_and_extensive_properties. It sounds like it has. But I'm not sure.
Peiyuan Zhu (Jun 09 2022 at 21:05):On this page "space and quantities" what does "probable" mean? https://ncatlab.org/nlab/show/space+and+quantity
John Baez (Jun 10 2022 at 02:07):Peiyuan Zhu said:
What are the pre-requisite[s] to understand cohesive topos[es]?
I'd say topos theory. There are a bunch of books on it listed here:
David Corfield (Jun 10 2022 at 10:45):Peiyuan Zhu said:
On this page "space and quantities" what does "probable" mean? https://ncatlab.org/nlab/show/space+and+quantity
The word written there is "probeable" or "probe-able", able to be probed. It's the idea of understanding some space by means of maps from some collection of simple test spaces, and how these maps then fit together. You can see this idea explained for graphs here.
David Corfield (Jun 10 2022 at 10:47):Peiyuan Zhu said:
I think it might have something to do with physics because Lawvere talks about intensive vs extensive quantities. I'm not sure if has anything to do with those in thermodynamics https://en.wikipedia.org/wiki/Intensive_and_extensive_properties. It sounds like it has. But I'm not sure.
This has been related to cohesion in the nLab section - intensive or extensive quantity: In modal homotopy type theory.
Peiyuan Zhu (Jun 15 2022 at 17:02):How to understand “judgement” in the sense of Per-Martin Löf? What problem is he trying to solve? I feel rather confused after reading his paper “Truth of a proposition…”
Peiyuan Zhu (Jun 15 2022 at 17:03):C0DA790D-7446-4D22-B91B-44CEB5C1427B.png
Peiyuan Zhu (Jun 15 2022 at 17:06):What do mathematicians value? In other words, what makes a theorem good? In my naive understanding it seems simplicity is good, and universality in the sense that it underlies intuition of many other proven ones is good, and it’s good because it solves practical problems. But I want to hear what @John Baez has to say.
John Baez (Jun 15 2022 at 18:05):It would take a long time to figure out what makes a theorem good - it's almost as hard as telling when a piece of music is good.
I believe there are many different kinds of good theorems, which is one reason why different mathematicians can have radically different styles.
Peiyuan Zhu (Jun 15 2022 at 18:33):Would it be possible to enumerate some of the "kinds" and "styles" that you're mentioning?
John Baez (Jun 15 2022 at 18:41):Well, just think about the theorems or conjectures you know, from Goldbach's conjecture to the formula for to the Yoneda embedding theorem, and you'll see that they have very different kinds of appeal.
John Baez (Jun 15 2022 at 18:42):I really don't want to get pulled into creating a taxonomy.
Naso (Jun 16 2022 at 00:49):@Peiyuan Zhu there is a podcast 'my favourite theorem' where they interview famous mathematicians about their favourite theorems: https://kpknudson.com/my-favorite-theorem
Paul Séjourné (Jun 17 2022 at 13:58):Hi a maybe naive question about terminology, I was reading Catégories Tannakienne (Tannakian category) from Neantro Saavedra Rivano, and I was then a bit confused about the 'post-modern' terminology. According to the Deligne article about the previous book Saavedra's ACU tensor-categories correspond to symmetric monoidal category. What about the differences between rigid category, autonomous category, *-autonomous category ? what are the links between closure property and rigidity property ? And finally, is there a link between the property from an objet to being rigid i.e. automorphism group is trivial, and the rigid terminology mention above ?
John Baez (Jun 17 2022 at 16:29):[[rigid monoidal category]] = [[autonomous monoidal category]] and [[star-autonomous category]] have pretty widely accepted definitions.
John Baez (Jun 17 2022 at 16:31):And finally, is there a link between the property from an objet to being rigid i.e. automorphism group is trivial, and the rigid terminology mention above ?
Not that I know of.
Paul Séjourné (Jun 17 2022 at 17:10):Another mistery, Saavedra, talking about associativity constraint, explain the following : " the group of automorphisms of the tensor functor act on the left on the set of associativity constraint. Two associativity constraint are then said to be cohomologous if they are in the same orbit for the previous action. " Did you ever seen this elsewhere ?
Paul Séjourné (Jun 17 2022 at 17:11):John Baez said:
[[rigid monoidal category]] = [[autonomous monoidal category]] and [[star-autonomous category]] have pretty widely accepted definitions.
Thank you.
Morgan Rogers (he/him) (Jun 18 2022 at 07:09):I haven't seen objects described as "cohomologous" before, and it's not at all clear to me that this has anything to go with being in the same cohomology class (as the term would suggest), but maybe I'm missing context on what exactly the "associativity constraints" are
Paul Séjourné (Jun 18 2022 at 08:49):I'm talking about associativity constraint for monoidal category, the one relate to the pentagon axiom. But I think this terminology of " cohomologous associativity constraint" is comming from Saavedra himself (or maybe from Grothendieck), but I could not find contemporary paper using it. I'm guessing that the reason is that this is a detail that does not lead to a result, the automorphism group of the tensor product is not appearing (as far as i know) anywhere in the rest of Saavedra's text.
John Baez (Jun 18 2022 at 23:16):Morgan Rogers (he/him) said:
I haven't seen objects described as "cohomologous" before, and it's not at all clear to me that this has anything to go with being in the same cohomology class (as the term would suggest), but maybe I'm missing context on what exactly the "associativity constraints" are.
"Associativity constraint" is a slightly old-fashioned term for the "associator" in a monoidal category. In the case of a monoidal category where the monoid of isomorphism classes of object is a group , we can describe associators as 3-cocycles in group cohomology. Namely, if is the automorphism group of the unit object, the associator can be massaged to give a function
and this is a 3-cocycle on with values in the -module .
John Baez (Jun 18 2022 at 23:16):Cohomologous choices of then give equivalent monoidal categories!
John Baez (Jun 18 2022 at 23:18):This was all worked out by Grothendieck's student Hoàng Xuân Sính, whom Wikipedia calls "the first female mathematician in Vietnam".
The cohomology class of is often called the "Sinh invariant". Her thesis on this is here.
John Baez (Jun 18 2022 at 23:19):I explained this as part of a much longer story here:
Paul Séjourné (Jun 19 2022 at 01:08):Wow incredible, thanks to you i'm not gonna sleep. The question is, how do I have not heard about her before. Completely incredible, thank you so much.
Paul Séjourné (Jun 19 2022 at 01:17):I'm still astonished
Peiyuan Zhu (Sep 21 2022 at 02:15):Does anyone have reference on how to study emergence? I found this paper of Joseph Goguen studying hierarchical organizations with category theory https://www.karger.com/Article/Abstract/389460 but I wonder if there's more
Simon Burton (Sep 21 2022 at 08:30):Google scholar has some suggestions: https://scholar.google.com/scholar?cites=4787239375252565982&as_sdt=2005&sciodt=0,5&hl=en
Matteo Capucci (he/him) (Sep 21 2022 at 09:08):I recommend Eli Adam's thesis, quite a good start
Peiyuan Zhu (Sep 21 2022 at 16:54):@Matteo Capucci (he/him) can you send a link?
Peiyuan Zhu (Sep 21 2022 at 16:54):@Simon Burton yeah I was looking at that too
Matteo Capucci (he/him) (Sep 21 2022 at 16:55):Peiyuan Zhu said:
Matteo Capucci (he/him) can you send a link?
I'm sure Google will yield one when you look up 'eli adam phd thesis'
Peiyuan Zhu (Sep 21 2022 at 17:00):Where do people get into complex system research? Some people told me about Santa Fe summer school, not sure if it’s worth it
Naso (Sep 25 2022 at 02:47):Is there a name for a presheaf equipped with a monoidal product on its category of elements?
Morgan Rogers (he/him) (Sep 26 2022 at 08:20):I don't know of one!
Joe Moeller (Sep 26 2022 at 13:22):This is a special case of a monoidal fibration.
Morgan Rogers (he/him) (Sep 26 2022 at 14:15):What @naso described seems more general than those situations, but maybe the case they were thinking about falls into one of those cases :grinning_face_with_smiling_eyes:
Joe Moeller (Sep 26 2022 at 15:03):Right, I just assumed the fibration would also be monoidal. It's feels mismatched to put a monoidal structure on the category of elements and not ask the fibration to be monoidal.
Naso (Sep 26 2022 at 23:58):So what is the monoidal fibration property in the case of a presheaf ? I'm mainly interested in the case when is the poset of opens of a topological space .
Joe Moeller (Sep 27 2022 at 15:00):If T is monoidal, and F is lax monoidal, there's a way to define a monoidal structure on . If is the laxator for F, x and y objects of T, a an object of Fx, b an object of Fy, then define the tensor on objects of by:
The projection onto the T component is a (split) monoidal fibration.
Joe Moeller (Sep 27 2022 at 15:08):Moreover, this gives an equivalence between the category of lax monoidal functors and the category of split monoidal fibrations over T.
Naso (Sep 29 2022 at 01:59):Do you mean that ?
So we need three monoidal categories , , and a lax monoidal functor (where , are arbitrary but is cartesian product?)
So the fibration being monoidal means
Yeah Morgan was right, I was assuming that should happen.
In there is a unique morphism iff .
( meaning restriction to )
Then the fact that is monoidal means if and , we should have
Is that right?
Joe Moeller (Sep 29 2022 at 13:19):Yeah, everything you said is right.
Naso (Sep 30 2022 at 03:31):Ok, it turns out for the I have in mind actually
doesn't hold but instead
holds (really is a presheaf of posets). Is there a way to understand this from the monoidal fibrations theory?
John Baez (Sep 30 2022 at 10:30):Peiyuan Zhu said:
Where do people get into complex system research? Some people told me about Santa Fe summer school, not sure if it’s worth it
It's probably "worth it" - here are 3 descriptions of what it's like.
Paul Séjourné (Dec 13 2022 at 14:50):Hi, surely a naive question : does anyone know if the Tannaka duality fits or not into a more general duality context, for example Isbell duality ? I know there may exist connections between them according to the Examples and similar dualities section of the Isbell duality nlab page but I'm struggling to find a clear answer.
More speculatively, is there any research/elaboration/notes/whatever on the philosophy of Tannaka duality ? except the one we can extract from Grothendieck thoughts.
Morgan Rogers (he/him) (Dec 13 2022 at 19:10):Hi Paul, I may be starting a project next year where I try to answer that kind of question using actegories (we can discuss the relevant/related results from my thesis whenever I next see you if you like). To me, the word "duality" in 'Tannaka duality' is somewhat misleading, since the original theorem of Tannaka is just a correspondence of objects (there's no adjunction involved a priori), and although the construction of the category of representations can be made functorial, there is no natural functor in the opposite direction until one has restricted the class of categories (and suitable functors). Based on the Wikipedia page, I think this naming convention comes from thinking about this is an extension of Pontryagin duality (where a compact abelian group is assigned the dual group of its characters).
On the other hand, one can relate the subjects of Tannaka duality to Isbell duality! We can construct an adjunction between the left representations of groups and right representations of groups (left and right modules for the group ring) by the usual dualization procedure sending a vector space to its dual , which inherits an action on the opposite side. This restricts to finite dimensional representations in particular.
Paul Séjourné (Dec 17 2022 at 14:00):I see your point about termonology, i think the duality in Tannaka duality refer to the reconstruction part of what i want to see as a more philosophical "duality" between reconstruction and recognition through tannakian formalism on which i'm working. I think I see what you mean about the relation between Isbell and Tannaka 'duality'. Do you think about transposing this kind of construction in the context of actegories ? am I understanding well ? Thank you for your answer i'll be glad to discuss it with you next time I come to the lab.
Peiyuan Zhu (Dec 30 2022 at 18:32):image.png
Here it says the identity is uniquely determined that communtes with any function composition, that applying it before, after is the same as not applying it. But I can think of a case when you don't have that many arrows in your category. You only have certain functions that composing the two does give you the identity, but none of which is the identity, like a matrix applied to its inverse gives you the identity, but there're only three linear maps in the category, that is the matrix itself, its inverse, and the identity. What am I missing? So is it saying all before and after has to hold at the same time, but not only one of them at a time? Because if it has to hold all at the same time, then only identity matrix satisfies this but not any of the invertible matrices. So maybe it's just my way of reading isn't "mathematician" enough, do everyone read that diagram as an AND connective instead of OR? Do mathematicians always assume they're reading the strongest voice to be put on the table?
Jean-Baptiste Vienney (Dec 30 2022 at 19:44):Hi Peiyuan, if you have exactly three morphisms , and , what about ? It can't be equal to the identity, because it will give you that and thus you will have at most two morphisms in your category: and ! When you think to a weird example, often it is that this example in fact doesn't exist!
Jason Erbele (Jan 22 2023 at 07:58):@Jean-Baptiste Vienney, there are a few categories with exactly three morphisms. There are categories with three morphisms and two objects that are not isomorphic to each other, and a category with three objects that has three (identity) morphisms. But if two of the three morphisms are non-identity morphisms, and , there can be only one object. While it is true in this case that , this does not exhaust what could equal. The other two options are itself and , and it's not hard to work out that exactly one of those two actually works.
To the question that @Peiyuan Zhu asked, the diagram in the image you posted is stated to be a commuting diagram, which means BOTH triangles AND the parallelogram encode equations that are satisfied in the category: AND AND . Generally speaking, definitions and theorems are like recipes. If a recipe gives you a list of ingredients, you need to use all of the ingredients unless an OR is explicitly stated.
Peiyuan Zhu (Feb 05 2023 at 08:05):This wikipedia page on adjoint functors https://en.wikipedia.org/wiki/Adjoint_functors says "In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories." Does "equivalence" here refers to equivalence of categories or isomorphism of categories? Does "weaker form" refers to existence of adjoint functors or isomorphisms of categories?
This blog on adjoint functors https://www.math3ma.com/blog/what-is-an-adjunction-part-1 says "We've just exchanged equalities = for isomorphisms ≅ , so what if we take this a step further and exchange the isomorphisms for regular morphisms?" so I suppose this is talking about the same thing but it takes two steps to weaken down isomorphism.
This wikipedia of Galois connection https://en.wikipedia.org/wiki/Galois_connection says ", a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets)" so when the underlying categories are posets adjunctions becomes Galois connections is that it? Are the natural isomorphism and order isomorphism two different things?
Also I don't think I get the idea of natural transformation / isomorphism illustrated here https://www.math3ma.com/blog/what-is-a-natural-transformation. It seems to describe a relationship between two functors. What does it mean when ηx is isomorphism? Is it talking about "iso arrow"? "F and G are really the same functor up to a change in perspective." sounds like something interesting and that makes sense in the sense of seeing "functors as analogies between processes" and the question when two analogies are the same by changing perspectives (am I getting this right?). Is the word "conjugation" in "So when each ηx is an isomorphism, the naturality condition is a bit like a conjugation!" referring to conjugate matrix? That quadratic form looks familiar but it doesn't look like it's talking about conjugate matrices?
Patrick Nicodemus (Feb 05 2023 at 10:26):It is a subjective claim that "adjunction is a weak form of equivalence." In my view it is simply untrue.
Tobias Schmude (Feb 05 2023 at 11:45):I'd call it a lax form of equivalence, and laxification tends to not preserve the flavor of things well.
Morgan Rogers (he/him) (Feb 05 2023 at 14:16):@Peiyuan Zhu re conjugation, it's referring to the concept from group theory, which starting from an element (or isomorphism) constructs the map . Here is substituted for the components of the natural isomorphism.
John Baez (Feb 05 2023 at 17:45):One way to rescue a theorem out of this vague idea is: every equivalence between categories gives an adjunction that is also an equivalence. The proof of this is pretty interesting.
Peiyuan Zhu (Feb 06 2023 at 01:09):image.png
I don't quite understand how " "sameness" is stricter than it need be". If both categories are power sets then what's the difference between isomorphism and natural isomorphism? Are they both https://en.wikipedia.org/wiki/Order_isomorphism?
Peiyuan Zhu (Feb 06 2023 at 01:43):The examples of isomorphism of categories / natural isomorphisms on these pages seem quite complicated. Are there easier examples?
https://en.wikipedia.org/wiki/Isomorphism_of_categories
https://en.wikipedia.org/wiki/Natural_transformation
Morgan Rogers (he/him) (Feb 06 2023 at 10:18):In a power set (or indeed in any partially ordered set), each isomorphism class contains only a single object/element, so any equivalence between such structures (viewed as categories) is an isomorphism. To give you some insight into equivalence, consider the equivalence between the category of finite-dimensional vector spaces over and the category whose objects are the natural numbers (where the number "represents" the vector space ) with morphisms being matrices with coefficients in . We can easily construct a functor from the latter category into the former by sending to and each matrix to the corresponding linear map. Conversely, using a strong form of the axiom of choice we can pick a basis for each finite dimensional vector space in order to produce a functor to the category with natural numbers as objects. This is an equivalence because every finite-dimensional vector space is isomorphic to for some , but not equal to it: there are some vector spaces which carry presentations that look very different to .
Peiyuan Zhu (Apr 28 2023 at 01:03):Is there a reason "adjoint functors arise everywhere"? Can I understand that as "counterfactual relationships arise everywhere"?
Peiyuan Zhu (Apr 28 2023 at 17:59):Is there any nice introductory book to "Game Semantics"? https://en.wikipedia.org/wiki/Game_semantics
Leopold Schlicht (Apr 28 2023 at 18:33):Peiyuan Zhu said:
Is there any nice introductory book to "Game Semantics"? https://en.wikipedia.org/wiki/Game_semantics
https://en.wikipedia.org/wiki/Game_semantics#Books
Peiyuan Zhu (Apr 28 2023 at 22:22):Most of those books are in German lol
Peiyuan Zhu (Apr 29 2023 at 00:06):Some a little too Advanced
Peiyuan Zhu (Nov 06 2023 at 17:37):Has there been any paper that says adjoint quantifiers are more suitable to study complex systems?
Peiyuan Zhu (Nov 06 2023 at 18:03):Or adjoint quantifiers as generalization of causal relations
Jencel Panic (Nov 11 2023 at 11:49):A good place to start when thinking about adjoint functors is to view them as a "weak form of equivalence" i.e. adjunction is when you have two categories that are not fully equivalent, but somewhat equivalent, typically because one is simpler than the other.
Todd Trimble (Nov 11 2023 at 11:55):Sorry, this "somewhat equivalent" doesn't accord with my intuition at all.
Adjoint functors are everywhere, and one should try to think of as many examples as possible, before speculating what they are generally like.
Jencel Panic (Nov 11 2023 at 12:04):Well, it is a popular starting point, especially if you want to understand quantifiers.
Todd Trimble (Nov 11 2023 at 12:56):Quantification does give examples of adjoint functors (quantification is adjoint to substitution, as Lawvere taught us long ago), but how is quantification somewhat like equivalence?
(This is mostly a rhetorical question.)
Todd Trimble (Nov 11 2023 at 22:50):I'll back off a little to say that I've certainly heard people try to explain adjunctions in similar ways. I can't repeat the explanation with much conviction myself -- it doesn't really "sing" to me to begin with -- but it's something on the order of "even if a functor isn't an equivalence, then having an adjoint is the next best thing".
The trouble I find is that it takes a lot of words, even for people who do find this sort of thing helpful or convincing, to explain what's really in their heads, why this makes adjoints so wonderful. Trying to keep it short and snappy at this level is, I think, bound to be misleading, give some wrong ideas. Hence my reaction.
In my own head, the power and real point of adjunctions is very bound up with the power of representable functors, and of universal properties, and of the Yoneda lemma... learning the yoga of these connections was pure gold for me. But it takes a while, and some persistent work on the part of the beginner, to learn it. Or at least I wish I knew a way to make it all seem smooth and easy.
Todd Trimble (Nov 11 2023 at 22:50):Peiyuan: sorry, "complex system" is just too vague for me to know how to deal with. Maybe I'm not the right person to respond, but is there a more specific case of what you have in mind that really grabs you?
Patrick Nicodemus (Nov 11 2023 at 23:02):Jencel Panic said:
Well, it is a popular starting point, especially if you want to understand quantifiers.
I agree with Todd that this is inadequate even as a starting point.
Patrick Nicodemus (Nov 11 2023 at 23:04):this is like saying that a homomorphism is a kind of weak isomorphism which need not be surjective or injective.
Patrick Nicodemus (Nov 11 2023 at 23:09):If I am asked to list examples of an adjunction for a beginner, I would say something like
For each of these examples it is difficult for me to understand it as a weak equivalence.
Patrick Nicodemus (Nov 11 2023 at 23:17):David Spivak characterizes adjunctions this way but I disagree with that.
He also describes universal properties as "generalizations of extrema and suprema from calculus"
which is again technically true but these are not the examples that have really driven or motivated the development of these notions.
Spencer Breiner (Nov 11 2023 at 23:43):With regards to adjunctions and equivalence, I think a more useful perspective is that an adjunction establishes an equivalence between data in one context and superficially different data in another
Spencer Breiner (Nov 11 2023 at 23:45):E.g., a function represented as pairs of input/output vs zero-one matrices
Peiyuan Zhu (Dec 22 2023 at 01:41):Is there any “universal determination” by adjoints?
Peiyuan Zhu (Dec 22 2023 at 01:47):Like the universal property in seven sketches
Peiyuan Zhu (Dec 22 2023 at 02:48):https://philpapers.org/archive/ELLAAE.pdf
Peiyuan Zhu (Dec 22 2023 at 02:57):In this paper it uses Cartesian product
Daniel Geisler (Dec 22 2023 at 03:42):Consider the relationship between the set partitions and the Bell polynomials. Is the functor that maps the set partitions to a Bell polynomials a natural transformation?
Peiyuan Zhu (Dec 31 2023 at 08:10):Daniel Geisler said:
Consider the relationship between the set partitions and the Bell polynomials. Is the functor that maps the set partitions to a Bell polynomials a natural transformation?
Let me try to prove this...
Peiyuan Zhu (Dec 31 2023 at 08:11):New question: What's the difference between propositional logic and modal logic? It looks like modal logic is really just quantifiers of "all possible worlds" and "at least one possible world"
Chris Grossack (they/them) (Dec 31 2023 at 14:04):Modal logic, broadly interpreted, is about interpreting logical connectives that are not truth functional. These are usually called modal connectives ormodalities, which explains the name.
Chris Grossack (they/them) (Dec 31 2023 at 14:09):For a definition by example, look at . This connective is truth functional, since the truth value of only cares about the truth of and the truth of . It doesn't matter what and actually represent.
Contrast this with a connective like , which we'll interpret as "I know". Then the truth value of depends on what represents, rather than just being a function of its truth value!
For instance, maybe "it's raining" and "it's Sunday" are both true. But "I know it's Sunday" can be true while "I know it's raining" is false (because I haven't stepped outside yet), so the knowledge modality is not truth functional.
Chris Grossack (they/them) (Dec 31 2023 at 14:13):It should come as no surprise that modalities show up VERY naturally when reasoning about real world systems. Modal logic is also "robustly decidable" in the sense that, for pretty much every modal logic you would think about, there's a program which will tell you whether a given sentence is true (and provide a proof) or false (and provide a counterexample).
The combination of applicability and decidability make modal logic an extremely useful tool, for instance in model checking.
Mike Shulman (Dec 31 2023 at 17:03):The connection of "possible worlds" is that in certain models of modal logic, some modalities such as "it is possible that" and "it is necessary that" can be interpreted as quantifications over "possible worlds". But the worlds don't exist in the syntax of modal logic; not all modalities can be interpreted that way (e.g. "I know that" doesn't have much to do with possible worlds); and even those that can be don't have to be.
James Deikun (Dec 31 2023 at 17:13):Well ... if I am logically omniscient, "I know that" can be interpreted as being about "worlds I could be in, as far as I know". If I am not logically omniscient, it's complicated to represent that and you can't use the standard sort of modal logic that has "robust decidability" and everything.
Mike Shulman (Dec 31 2023 at 17:16):Okay, maybe you can pick holes in the one example I used (which I picked because it's the one Chris used), but there are lots of modalities that clearly don't have anything to do with possible worlds, like "it holds discontinuously that".
James Deikun (Dec 31 2023 at 17:19):But can't any modal logic that admits Kripke-style semantics be interpreted as being about worlds? It's just a question of how natural it is.
Mike Shulman (Dec 31 2023 at 17:57):Sure, but that's sort of a tautologous statement: "Kripke-style semantics" is basically by definition about worlds.
Evan Washington (Jan 01 2024 at 04:55):It's important to note that a 'world' in Kripke semantics is just whatever sort of thing should be interpreted as a point of evaluation in the application/flavor of modality at issue. It could be an information state or an epistemic possibility or a partial world or memory state or a time or whatever.
Here's a fun example (which I believe is due to van Benthem): first-order logic is a (multi) modal logic, with modal operators for each variable . Here the 'worlds' are assignment functions (and the -variant relation as accessibility).
Evan Washington (Jan 01 2024 at 05:08):Peiyuan Zhu said:
New question: What's the difference between propositional logic and modal logic? It looks like modal logic is really just quantifiers of "all possible worlds" and "at least one possible world"
There are many species of modal logic. Propositional modal logic is typically a propositional logic with some non-truth-functional connectives . On the interpretation given by Kripke semantics, those do correspond to (restricted) quantifications over possible worlds. But (a) modal logics existed long before that interpretation was ever given (that's why Kripke's completeness result was a big deal) and (b) not all modal logics admit such an interpretation.
Peiyuan Zhu (Jan 03 2024 at 06:23):In MLTT is it possible to make modal judgement like or "?
James Deikun (Jan 03 2024 at 09:30):MLTT as such doesn't have modal connectives, however there are extensions and variations of it that do. I believe HTT is one such.
Peiyuan Zhu (Jan 03 2024 at 20:18):Can you geive a pointer to this specifically?
Peiyuan Zhu (Jan 05 2024 at 03:27):In HTT
Kevin Arlin (Jan 05 2024 at 18:00):I suspect James meant H(o)TT, not Higher Topos Theory. A lot of stuff on modalities in homotopy type theory should be accessible starting from the nLab since it's a particular interest of Urs's.
Mike Shulman (Jan 05 2024 at 18:23):Book HoTT does not really have modal logic of this sort. You can construct inside of it certain monadic modalities, but it doesn't really support modal logic of the sort that modal logicians mean. Recently some general modal dependent type theories have been introduced, such as MTT.
Peiyuan Zhu (Jan 19 2024 at 02:34):@John Baez Is there any book you recommend on how to understand the notion of "causality" in physics and statistics?
Peiyuan Zhu (Apr 27 2024 at 18:52):by the computational trinitarianism, what is the equivalence of homotopy invariance in proofs?
Peiyuan Zhu (Apr 27 2024 at 19:29):Is it then about equivalences of proofs, that they start from the same premises and end in the same conclusions?