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Stream: theory: category theory

Topic: introduction to categories as algebraic gadgets

Mike Shulman (May 16 2023 at 15:33):

Can anyone recommend an introduction to category theory that emphasizes the perspective of categories as a kind of algebraic structure, generalizing groups, monoids, and posets? This perspective is of course present to varying degrees in all introductions; but I'm looking for something that would be maximally accessible to a student with limited background in abstract mathematics, so that many of the "usual" examples of large categories would not be familiar.

David Egolf (May 16 2023 at 15:53):

As someone with a limited background in abstract mathematics, I've found parts of these two fairly accessible:

• "Notes on Category Theory with examples from basic mathematics" by Perrone
• "Seven Sketches in Compositionality: An Invitation to Applied Category Theory" by Fong and Spivak

Parts of these books talk about categories in the way you are describing.

Also, the very first book on categories I looked at was "Category Theory for Scientists" by Spivak. This book did a great job at making category theory interesting to me, although the emphasis on "ologs" isn't to my preference.

I haven't looked carefully at this book, but "The Joy of Abstraction" by Cheng I think has a few sections focused on examples along the lines you describe above.

I should also mention "Algebra: Chapter Zero" by Aluffi. I find it hard reading, but very rewarding. I'm not sure I would call it "maximally accessible", but it does talk a bit about categories in the context of algebra.

John Baez (May 16 2023 at 16:27):

My category theory lecture notes has a section called "Categories as mathematical objects" on page 6, which is about algebraic gadgets like posets and monoids as categories. It's very terse and skeletal, with typos, since some students created these notes and I haven't yet gotten around to polishing them. Someday I hope it will do more of what you want, @Mike Shulman.

John Baez (May 16 2023 at 16:33):

But I want this book to ultimately play off the two perspectives against each other: "categories of mathematical objects" and "categories as mathematical objects".

David Egolf (May 16 2023 at 18:10):

John Baez said:

But I want this book to ultimately play off the two perspectives against each other: "categories of mathematical objects" and "categories as mathematical objects".

That sounds interesting!
By the way, I'd also be interested in the perspective "mathematical objects as categories", which would presumably involve discussing different strategies for constructing categories from mathematical objects, and then studying those objects from the resulting categories.

Leopold Schlicht (May 16 2023 at 19:49):

I like Part 0 of Lambek--Scott. It's a very efficient 40-page summary of basic category theory, organized by "slogans". Slogan II says that many mathematical objects -- such as sets, monoids, and preordered sets -- can be viewed as categories.

Leopold Schlicht (May 16 2023 at 20:30):

David Egolf said:

By the way, I'd also be interested in the perspective "mathematical objects as categories", which would presumably involve discussing different strategies for constructing categories from mathematical objects, and then studying those objects from the resulting categories.

There are a lot of results in that direction in various subfields of mathematics: Tannaka--Krein duality, Gabriel--Rosenberg reconstruction, Makkai duality, Grothendieck's Galois theory, ...

Leopold Schlicht (May 16 2023 at 20:48):

@David Egolf As far as I know, the idea of associating to a space its category of sheaves also played a crucial role in the construction of étale cohomology, because it allowed one to use the formalism of derived functors. This in turn was used in the proof of the Weil conjectures. I don't know much about it, but it feels like this is the perfect example of category theory being used in mathematics to prove concrete statements about the number of solutions of polynomial equations (over finite fields).

David Egolf (May 16 2023 at 20:54):

Thanks @Leopold Schlicht! That's interesting to hear about! Most of the things you've just mentioned go largely over my head, but it's great to have a list of examples I can return to as I learn more.

Leopold Schlicht (May 16 2023 at 21:02):

They mostly go over my head as well. :sweat_smile: All I know is that your idea of "constructing categories from mathematical objects, and then studying those objects from the resulting categories" is used quite heavily in mathematics, and several people got Fields medals for exploiting it. :grinning_face_with_smiling_eyes:

John Baez (May 17 2023 at 00:30):

David Egolf said:

John Baez said:

But I want this book to ultimately play off the two perspectives against each other: "categories of mathematical objects" and "categories as mathematical objects".

That sounds interesting!
By the way, I'd also be interested in the perspective "mathematical objects as categories", which would presumably involve discussing different strategies for constructing categories from mathematical objects, and then studying those objects from the resulting categories.

That's what "categories as mathematical objects" was supposed to be about, actually. Maybe I should say it the other way. My idea, anyway, was to get people used to thinking of categories not only as "universes" like "the category of all groups" or "the category of all posets", but also as small manageable things just like individual groups and posets and equivalence relations... in part by noticing that groups and posets and equivalence relations are categories!

John Baez (May 17 2023 at 00:36):

Leopold Schlicht said:

There are a lot of results in that direction in various subfields of mathematics: Tannaka--Krein duality, Gabriel--Rosenberg reconstruction, Makkai duality, Grothendieck's Galois theory, ...

These things are great, but some of them don't belong in a basic introduction to category theory, because they have significant prerequisites from pure math, which tends to intimidate a lot of people, while attracting others.

It would, however, be great to write a series of books teaching math with categories, starting with basic stuff and leading up to these fancier examples.

John Baez (May 17 2023 at 00:38):

If that could be done right, it could demystify a bunch of these topics.

Mike Shulman (May 17 2023 at 00:48):

I think the phrase "categories as mathematical objects" is potentially confusing, because large categories such as "the category of all groups" are also mathematical objects!

Mike Shulman (May 17 2023 at 00:48):

But it sounds as if you intend that phrase to mean something like "categories as small algebraic gadgets".

Evan Patterson (May 17 2023 at 00:58):

I have gotten in the habit of saying "category theory in the small" vs "category theory in the large" to mark this distinction, especially in cases where both are going on at the same time. I forget where I picked up that phrase (although obviously it is inspired by the technical distinction between small and large categories).

Category theory in the small tends to be grossly underappreciated!

David Egolf (May 17 2023 at 01:03):

I have seen several introductory books that talk about modelling a preorder, group, or monoid as a category (as mentioned above). However, I don't remember seeing an introductory book that talks about a ring as a category, or a topological space as a category. In a hypothetical book that talks about how certain mathematical structures can be easily modelled as categories, it might be nice to also give some non-examples: mathematical objects that take a bit more work (or more fancy notions) to describe as categories (or perhaps are not productive to try and view as categories). To the beginning reader (like me!), it would be instructive to learn why some mathemtical objects are not as easily modelled as categories.

Mike Shulman (May 17 2023 at 01:12):

I think the examples of algebraic structures like posets, groups, and monoids that can literally be viewed as categories are very special. In general, there's no reason an arbitrary mathematical structure should be modelable as a category. To me the viewpoint of "category theory in the small" is just that categories are an algebraic structure like groups, monoids, rings, etc.

John Baez (May 17 2023 at 05:27):

Mike Shulman said:

But it sounds as if you intend that phrase to mean something like "categories as small algebraic gadgets".

I don't really think small/large in the set-theoretic sense is the essence of the distinction I'm trying to make, though it tends to be correlated to it. When introducing category theory, I don't want my readers to think at all about size issues in the set-theoretic sense. But I do want to point out the difference between using a category as a context in which you study objects of some sort, versus treating a category itself as the object of study. This distinction is psychologically important to beginners. Too many introductions to category theory focus on the first kind of example, not enough on the second. As a result, they tend to visualize a category as a misty sort of "background" in which objects sit. I want them to also be able to see a category as something like a tiny ant.

John Baez (May 17 2023 at 05:31):

David wrote:

However, I don't remember seeing an introductory book that talks about a ring as a category, or a topological space as a category.

Yes, these examples are a bit more sophisticated. But they're really important! So I think a good modern intro to categories has got to talk about enriched categories at some point, so readers can see a ring as a one-object Ab-enriched category. And I think it's got to talk a bit about locales, so readers can learn to think of topology not in terms of a set of points with a topology has slapped on, but rather a poset of "opens".

John Baez (May 17 2023 at 05:33):

My lecture notes didn't get that far, and I'm rather intimidated by the amount of work required to write a really good introduction to category theory!

Mike Shulman (May 17 2023 at 05:46):

Yeah, I didn't necessarily mean "small" in the technical set-theoretic sense; I meant it in the same sense you did when you mentioned "small manageable things just like individual groups and posets".

John Baez (May 17 2023 at 05:47):

Okay, good!

John Baez (May 17 2023 at 05:50):

Part of why I want to encourage beginners to think a lot about "small, manageable" categories is that they can then transfer the some of the intuitions they build that way to large categories like Set and Grp, or at least lose some of their fear of these these categories.

John Baez (May 17 2023 at 05:51):

I remember once I was talking to Ross Street and I called a bicategory $x$ and he was very impressed - a lot of people, at least back then, would denote a bicategory with some really fancy calligraphic capital letter because it's even bigger and scarier than a category!

Leopold Schlicht (May 17 2023 at 10:05):

John Baez said:

Leopold Schlicht said:

There are a lot of results in that direction in various subfields of mathematics: Tannaka--Krein duality, Gabriel--Rosenberg reconstruction, Makkai duality, Grothendieck's Galois theory, ...

These things are great, but some of them don't belong in a basic introduction to category theory, because they have significant prerequisites from pure math, which tends to intimidate a lot of people, while attracting others.

It would, however, be great to write a series of books teaching math with categories, starting with basic stuff and leading up to these fancier examples.

I agree. But I do think a great introduction to category theory should at least sketch some proper applications of category theory to mathematics going beyond "category theory is a cool unifying language". I like the examples given in this MSE answer: https://math.stackexchange.com/questions/29152/motivation-and-use-for-category-theory/29159#29159

John Baez (May 17 2023 at 16:04):

My dream book would teach kids lots of pure math using category theory; they wouldn't need to know this math ahead of time, and everything would be a lot faster and easier and clearer and more unified and more fun than in a traditional treatment. So I wouldn't be focused on people who say "I already know math, how can category theory help me prove things I can't already prove?" It would be focused on people who say "I want to learn lots of math. Can you teach me?"

John Baez (May 17 2023 at 16:10):

Eugenia Cheng's book does a great job of this for people who have no experience of pure math at all: it starts as a course on what pure math is all about, explains things like posets and equivalence relations, then explains categories and leads up to the Yoneda embedding theorem.

John Baez (May 17 2023 at 16:13):

Tai-Danae Bradley, Tyler Bryson and John Terilla's book Topology with Categories teaches topology using category theory.

John Baez (May 17 2023 at 16:15):

But my dream book would teach group theory, ring theory, poset theory and topology, treating them as parts of category theory, while teaching the category theory.

John Baez (May 17 2023 at 16:15):

This would be a hard book to write!

John Baez (May 17 2023 at 16:21):

An easier book, aimed at 2nd-year math grad students, would assume they knew a bit of group theory, ring theory, poset theory and topology, and show how these are all category theory. It wouldn't just study categories of these things, it would show that these things are categories, and use that to do cool stuff. Monoids, groups, posets and locales are categories, rigs and rings are enriched categories!

Leopold Schlicht (May 17 2023 at 16:43):

John Baez said:

David Egolf said:

John Baez said:

But I want this book to ultimately play off the two perspectives against each other: "categories of mathematical objects" and "categories as mathematical objects".

That sounds interesting!
By the way, I'd also be interested in the perspective "mathematical objects as categories", which would presumably involve discussing different strategies for constructing categories from mathematical objects, and then studying those objects from the resulting categories.

That's what "categories as mathematical objects" was supposed to be about, actually. Maybe I should say it the other way. My idea, anyway, was to get people used to thinking of categories not only as "universes" like "the category of all groups" or "the category of all posets", but also as small manageable things just like individual groups and posets and equivalence relations... in part by noticing that groups and posets and equivalence relations are categories!

I think @David Egolf had something different in mind: You have a mathematical object (such as a scheme $X$), assign to it a category in some way (such as the étale topos of $X$), and use this category to study the mathematical object (such as: defining the étale cohomology of $X$ as the cohomology of the étale topos of $X$ -- and then using this invariant to prove things). This is different than the "categories as small manageable things like monoids and posets" perspective you are talking about.

Leopold Schlicht (May 17 2023 at 16:50):

By the way, my example combines both the "categories as large universes" and the "categories as small objects" perspective: topoi are "large universes" of sheaves, but Grothendieck emphasized that they should be thought of as small geometric objects.

John Baez (May 17 2023 at 16:51):

We'll have to ask @David Egolf what he was thinking, but I'm pretty sure he wasn't thinking about schemes and étale cohomology.

Leopold Schlicht (May 17 2023 at 16:53):

I'm not saying he was talking about schemes and topoi. But I think he was talking about assigning to an ordinary mathematical object a category and then studying the object via this category, and not literally viewing the object as a category. I came up with the example of schemes and topoi to explain to you what kind of thing I think he had in mind.

John Baez (May 17 2023 at 16:56):

Okay. Anyway, I'm curious what examples he had in mind.

Leopold Schlicht (May 17 2023 at 17:04):

I don't think he necessarily had a specific example in mind. Rather, I think he was just curious if there are cases in mathematics in which one studies a mathematical object by assigning to it a category. But we'll let him talk for himself. :grinning_face_with_smiling_eyes:

David Egolf (May 17 2023 at 18:48):

Well, all the stuff being discussed in this thread is quite interesting!

But here's what I had in mind:

Assume we have some mathematical object of interest. I'm interested in the process of generating categories that are helpful for studying it. This might look like viewing the mathematical object itself as a category, but it could instead involve viewing the mathematical object as an object in some category, or as a functor between categories. This could also involve studying the relationships between similar mathematical objects, when each is modelled as an object in some category. Broadly, I'm interested in the process of producing statements in the language of category theory that tell us things about a mathematical object.

One reason why I think this can be difficult is that a lot of flexibility is available. For each category we deem useful to studying a mathematical object of interest, we can produce many new categories and functors - each of which potentially provide a slightly different angle of study. Learning about examples of how people have navigated this flexibility to produce categories useful for studying the original mathematical objects would be interesting to me.

Closely related to this, I'm also interested in studying the process of mathematical study, using the language of category theory. It's awesome how category theory can help us draw parallels between theorems in different categories about mathematical objects, but - for example - it would also be exciting to use it to draw parallels between strategies used for constructing or defining interesting objects. To take a particular example, there are quite a few concepts that come together to define a smooth manifold - could one tease apart the process of reasoning that puts all these concepts together, and then state it in more general category theoretic language that could be applied in other settings?

John Baez (May 17 2023 at 19:06):

Okay, thanks! Those are very broad goals, and it would be nice if some mathematician thought about them and tried to discuss them at this level of breadth. I haven't seen anything except dozens of special cases.

John Baez (May 17 2023 at 19:08):

As for the concept of "smooth manifold" in particular, it's worth noting that this concept is built using a pattern that's used elsewhere, e.g. in defining [[topological manifolds]], [[piecewise-linear manifolds]], $C^k$ manifolds, analytic manifolds, [[complex manifolds]], and other things mathematicians like to think about.

John Baez (May 17 2023 at 19:09):

The nLab article [[manifold]] discusses a couple of attempts to formalize this pattern and come up with a general concept that specializes to all these concepts.

John Baez (May 17 2023 at 19:14):

However, there is roughly 1000 times as much work on each of the various kinds of manifolds I listed than on the attempt to build a general theory!

John Baez (May 17 2023 at 19:26):

Each individual kind of manifold has a research community that studies that kind and its relations to the other kinds.

John Baez (May 17 2023 at 19:30):

There's an interesting problem with all these kinds of manifolds. Each kind of manifold is a kind of space where locally you have a clear picture of what that space is like. This makes it fairly easy to work locally on manifolds, so a lot of the deeper questions involve the global study of manifolds.

John Baez (May 17 2023 at 19:32):

But the presence of this clear and quite restrictive local description of manifolds means that the category of manifolds of any particular kind is quite lacking in limits and colimits. They always tend to have binary products and coproducts, but never have all equalizers and coequalizers.

John Baez (May 17 2023 at 19:33):

This is a real problem, so people who want to use category theory to work with manifolds often get pushed away toward other formalisms, which categories with more limits and colimits.

John Baez (May 17 2023 at 19:34):

This is why people like [[diffeological spaces]] and [[differentiable stacks]].

John Baez (May 17 2023 at 19:35):

For example, I was forced into using diffeological spaces in my work on physics.

Jean-Baptiste Vienney (May 17 2023 at 19:49):

There is another kind of things that you can code as categories than algebraic structures, it is spaces. If you take a topological space, you can look at the categories whose objects are points of the topological spaces and a morphism $a \rightarrow b$ are continuous functions $f:[0,1] \rightarrow X$ such that $f(0)=a$ and $f(1)=b$ and you can compose them.

Well, you can do some higher-categorical homotopical stuff starting with this.

But one day, I wanted to use the idea for other type of spaces with points but where paths can be some other stuff. And another example that can come to your mind is affine spaces. Now the paths are vectors between points and you can still compose them.

I tried to generalize this to give a definition of something that I wanted to call affine categories which would code more fancy type of spaces than affine spaces but which are still affine in the sense that all the paths between $a$ and $b$ are the same than the path between $a+f$ and $b+f$ (defining things in a way that this notation make sense), for example the paths on a grid or some other stuff . I think that even some topological spaces with the previous type of categories would be "affine categories", maybe the uniform spaces. I tried to explain this stuff to people on Zulip in order than they play with me but at the time nobody wanted to play with me. Maybe I should try to re-explain my idea. I didn't do anything with that because I already have some other stuff to play with during my days. I don't even remember what was my definition but I wrote it somewhere on this Zulip (in a topic like "open research")

Jean-Baptiste Vienney (May 17 2023 at 19:53):

You can also see some board games as kind of categories where the positions are the objects and the authorized moves are morphisms if you want fun categories easy to understand (I'm not so sure with this but I heard some talk with that kind of categories one time I think, by Peter Selinger)

Jean-Baptiste Vienney (May 17 2023 at 20:09):

There are also the databases as categories that some dudes like (and make money with)

Jean-Baptiste Vienney (May 17 2023 at 20:50):

(who are they :face_in_clouds:)

John Baez (May 17 2023 at 21:01):

You're probably talking about @David Spivak and @Ryan Wisnesky.

Jean-Baptiste Vienney (May 17 2023 at 21:03):

Some beginners try to help a bit too

Jean-Baptiste Vienney (May 17 2023 at 21:05):

I mean I don't want to lack of professionalism, I don't know what I can say :sweat_smile:

David Egolf (May 17 2023 at 22:23):

Thanks to both of you for your interesting responses! It's always fun to hear about concepts completely new to me - I had never heard of "diffeological spaces" before, for example.

However, to appreciate these fancier concepts, I suspect I would need to learn a lot more about the historical concepts they emerged from. There is so much to learn!

John Baez (May 18 2023 at 00:05):

Diffeological spaces are less fancy than manifolds, by the way. That's why they form a nicer category. But they are newer, for sure.

John Baez (May 18 2023 at 00:10):

A diffeological space is a set $X$ and for each $n$ a set of maps $f: \mathbb{R}^n \to X$ called plots. These sets of plots need to obey 3 axioms that are very believable if we want to think of these plots as smooth maps from $\mathbb{R}^n$ to $X$ - and once we define "smooth maps" between diffeological spaces, that's exactly what these plots are!

John Baez (May 18 2023 at 00:12):

The reason this is less fancy than a manifold is that we don't start by requiring $X$ to be a topological space, so we don't introduce the concept of an atlas of charts that involves an open cover of $X$, and we don't need a concept of "maximal atlas".

John Baez (May 18 2023 at 00:17):

For anyone who wants to know - I know you're probably overloaded with incoming data, @David Egolf - the Wikipedia article gives the definition quite simply. It uses arbitrary open subsets of $\mathbb{R}^n$'s where I was just using $\mathbb{R}^n$'s. The two approaches are equivalent, but probably theirs looks simpler when you're first getting started.

Leopold Schlicht (May 18 2023 at 19:59):

David Egolf said (emphasis mine):

Well, all the stuff being discussed in this thread is quite interesting!

But here's what I had in mind:

Assume we have some mathematical object of interest. I'm interested in the process of generating categories that are helpful for studying it. This might look like (...), but it could instead involve viewing the mathematical object as (...), or as a functor between categories.

By the way, as you probably know, the Yoneda lemma basically says that any object, in any category $C$, can be viewed as a contravariant functor $C \to \mathbf{Set}$.

John Baez (May 18 2023 at 20:42):

And, just to balance things out, the coYoneda lemma says any object of $C$ can be viewed as a covariant functor $C \to \mathbf{Set}$.

Frank Tsai (May 18 2023 at 20:48):

Carlo has 2 YouTube videos on this topic. A contravariant functor is also called a presheaf.
https://www.youtube.com/watch?v=CY21lm4DfJE

Mike Shulman (May 19 2023 at 03:51):

John Baez said:

And, just to balance things out, the coYoneda lemma says any object of $C$ can be viewed as a covariant functor $C \to \mathbf{Set}$.

Some people (including me) say that that's just another version of the Yoneda lemma (applied to $C^{\rm op}$); whereas the [[coYoneda lemma]] says that every presheaf is a colimit of representables.

Mike Shulman (May 19 2023 at 04:20):

(In particular, technically the thing that can be viewed as a covariant functor $C\to \rm Set$ is an object of $C^{\rm op}$, not an object of $C$. Of course $C$ and $C^{\rm op}$ have the same objects, but the point is that the natural transformations between such covariant functors correspond to morphisms in $C^{\rm op}$ rather than in $C$.)

Josselin Poiret (May 19 2023 at 07:43):

Mike Shulman said:

Some people (including me) say that that's just another version of the Yoneda lemma (applied to $C^{\rm op}$); whereas the [[coYoneda lemma]] says that every presheaf is a colimit of representables.

the naming is very unfortunate though, esp. when you consider that this is an instance of "every X is a colimit of free X" which generally doesn't have a name. I'm a partisan of talking about the Yoneda "principle" instead, and then all other results are corollaries of this principle.

Josselin Poiret (May 19 2023 at 07:47):

ie. for a fixed $C$, presheaves on $C$ are algebras for a multi-sorted theory defined by $C$, where you have 1-ary operators between sorts and some equalities between them. Then you can view $y(c)$ as the free algebra generated by one element at $c$, and freeness here means that you have an adjunction given by a right adjoint $Psh(C) → Set$, the evaluation at $c$ (restriction to the $c$ sort) and the left adjoint given by the free algebra generated by a set at $c$.

Josselin Poiret (May 19 2023 at 07:48):

now saying that these multi-sorted algebras are "generated" by their elements at all sorts seems very intuitive, right?

Leopold Schlicht (May 19 2023 at 13:32):

Another term would be "density theorem".

Mike Shulman (May 19 2023 at 14:02):

Why is it unfortunate to give a special name to a specially important instance of something?

Josselin Poiret (May 21 2023 at 08:27):

Mike Shulman said:

Why is it unfortunate to give a special name to a specially important instance of something?

A lot of people who encounter this theorem view it as a magical hammer rather than something very intuitive, and I think the name contributes to that feeling. Also, the "co" in co-Yoneda doesn't make much sense to me. It just feels like an attempt to bend the Yoneda name to encompass every fact related to presheaves. Just give it a different name!

Mike Shulman (May 21 2023 at 14:42):

When doing (co)end calculus, it behaves very much like a dual of Yoneda. Yoneda is $\int_{c\in C} \hom(C(c,x),Fc) = Fx$, coYoneda is $\int^{c\in C} C(x,c) \times Fc = Fx$.

Josselin Poiret (May 21 2023 at 15:58):

Mike Shulman said:

When doing (co)end calculus, it behaves very much like a dual of Yoneda. Yoneda is $\int_{c\in C} \hom(C(c,x),Fc) = Fx$, coYoneda is $\int^{c\in C} C(x,c) \times Fc = Fx$.

ah, when viewed that way I can understand why. I like that statement much better too, very concise :)