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Stream: community: discussion

Topic: Open problems in category theory

Andrius Kulikauskas (Nov 23 2024 at 16:53):

I think that formulating and sharing a conjecture may sometimes require an environment where it is acceptable to think outside of the community's current language. So, for example, I became interested in the Yoneda lemma because of the "Yoneda property" that you can understand an object by way of its relationships with other objects. Metaphysically, in my understanding, this relates to knowledge Why. And, in my understanding, there are four levels of knowledge: Whether, What, How, Why. So I conjectured that all four levels appear in the Yoneda Lemma. Four years later that led to this presentation The Yoneda Embedding Expresses Whether, What, How, Why and the video that I made, with 8,000 viewers so far. I include this example here in my Theory Translator, along with about 40 similar examples of these four levels of knowledge.

I have another conjecture which is that it is possible to classify adjunctions in a metaphysically significant way. I have been collecting examples of adjunctions. I am happy with the preliminary six-fold classification that I arrived at. But it would be better to work on this with others. Perhaps there is a way to formulate this in terms that would interest mathematicians in academia.
ClassificationOfAdjunctions2.png

If you look at the two results, then you will see they are connected, as the four levels of knowledge appear among the six kinds of adjunctions. I have my own conjecture regarding that and may explore that in the future. Sometimes it's better to work alone or to organize one's own community. But it's good if communities can overlap.

Notification Bot (Nov 23 2024 at 17:23):

7 messages were moved here from #community: discussion > Attitudes of mathematicians by Morgan Rogers (he/him).

Jean-Baptiste Vienney (Nov 23 2024 at 17:26):

Can you phrase your classification in mathematical terms?

Jean-Baptiste Vienney (Nov 23 2024 at 17:29):

Once you have written a mathematical proposition, we can evaluate whether it seems true or not and whether we know how to prove it or not.

Jean-Baptiste Vienney (Nov 23 2024 at 17:30):

To sum up, people would like:
1) A mathematical proposition
2) which seems true,
3) which seems to require some effort to prove or disprove.

Jean-Baptiste Vienney (Nov 23 2024 at 17:32):

If you don't know how to formulate a precise mathematical proposition, at least it should be almost a precise mathematical proposition.

Andrius Kulikauskas (Nov 23 2024 at 18:43):

@Jean-Baptiste Vienney I think that was what I was trying to say with my initial sentence:

I think that formulating and sharing a conjecture may sometimes require an environment where it is acceptable to think outside of the community's current language

You ask, "Can you phrase your classification in mathematical terms?"

We can ask, what do you mean by "classification"? How would you phrase that with mathematical terms?

I think it becomes an interesting conversation, what kind of classification is meaningful? And then we could apply that to adjunctions. What would it mean to classify them? I gave a picture with six kinds. I am not aware of any other kinds of adjunctions so I appreciate additional kinds very much.

Jean-Baptiste Vienney (Nov 23 2024 at 18:49):

There is not a mathematical definition of "classification" but here it would be expected that you use some category-theoretic language such as functors, natural transformations etc...

fosco (Nov 23 2024 at 20:11):

We can ask, what do you mean by "classification"?

I can try to give a definition...

Given a setoid $(S,\rho)$, which means a set with an equivalence relation that allows one to identify equivalent elements, a "classification" consists of a $\rho$-transversal subset of $S$, i.e. a $T\subseteq S$ with the property that the intersection $T\cap U_\rho$ has exactly one element, for every element $U_\rho$ of the partition of $S$ associated to $\rho$.

It essentially means you're taking one and only one element for each equivalence class.

Usually, a classification problem is stated in terms of the relation of being isomorphic on a class of structures.

fosco (Nov 23 2024 at 20:12):

(for example: give me a distinguished representative for every isomorphism class of finite groups)

Alex Kreitzberg (Nov 23 2024 at 20:21):

Yeah I think of classifications similarly, my intuition is you have a seemingly "big" bunch of stuff $X$, and a "small" bunch of names $Y$, then a classification is an $f : X \rightarrow Y$, such that $f(x) \cong f(x')$ implies $x$ and $x'$ are "the same".

So, for example, if two "nice" spaces have the same number of holes (their genus) then they're "the same".

But I'd be surprised if all classifications fit this pattern, even in the (very) loose way I've stated the above.

Alex Kreitzberg (Nov 23 2024 at 20:27):

The easier $Y$ is to understand, the more useful the classification, but "easier to understand" can't really be given a careful definition. So I think that makes giving a careful definition of something as broad as "classification" risky.

Alex Kreitzberg (Nov 23 2024 at 20:28):

But a specific classification should be carefully definable.

Peva Blanchard (Nov 24 2024 at 00:15):

I know two instances of classifications that worked quite well:

• phylogenetic trees to classify living species
• Mendeleev's periodic table

One peculiar feature of both, if I remember correctly, is that they allowed to make predictions ("there must be something like X, Y, etc. out there") that turned out to be correct. The extreme anti-example would be to put a random label on everything and say "here's my classification": that wouldn't be really helpful.

John Baez (Nov 24 2024 at 00:22):

Classifications in math are a bit different. Among the most famous are the classifications of finite fields, finite abelian groups, simple Lie algebras over the complex numbers (harder), and finite simple groups (vastly harder).

These all fit into @fosco's general framework: we've got a category, and we describe a representative of each isomorphism class.

John Baez (Nov 24 2024 at 00:33):

Here's a very nice classification problem in category theory: classify all Lawvere theories for which all finitely generated models are free. There turn out to be 6 kinds. Instead of describing the Lawvere theories syntactically, I'll just say what their models are:

• sets
• pointed sets
• sets where all elements are equal
• pointed sets where all elements are equal
• affine spaces over a chosen division ring
• pointed affine spaces over a chosen division ring

The last one is usually described as "modules over a chosen division ring", and if the division ring is commutative they're usually called "vector spaces over a chosen field". But I wanted to emphasize the parallels.

This classification theorem is apparently not at all trivial.

David Egolf (Nov 24 2024 at 01:35):

If wonder if anyone has dreamed up a category where the objects are adjunctions. I'm not sure what a morphism between adjunctions should be like! But such a category would presumably provide a framework for classifying adjunctions - you could study the isomorphism classes.

David Michael Roberts (Nov 24 2024 at 02:02):

Well, yes, there is a category where the objects are adjunctions. If the morphisms are natural isomorphisms between one of the functors, then this category is rather boring. But if you take say natural transformations between one of the functors, this will be less trivial.

John Baez (Nov 24 2024 at 03:37):

I guess there's a double category where the objects are categories, the vertical morphisms (say) are functors, the horizontal morphisms are functors and the adjunctions are natural transformations. But in a way you're then just taking a 2-category (Cat) and building a double category which contains all the information in that 2-category, where the vercial morphisms are morphisms in that 2-category and the horizontal ones are adjunctions.

Jean-Baptiste Vienney (Nov 24 2024 at 04:55):

fosco said:

We can ask, what do you mean by "classification"?

I can try to give a definition...

Given a setoid $(S,\rho)$, which means a set with an equivalence relation that allows one to identify equivalent elements, a "classification" consists of a $\rho$-transversal subset of $S$, i.e. a $T\subseteq S$ with the property that the intersection $T\cap U_\rho$ has exactly one element, for every element $U_\rho$ of the partition of $S$ associated to $\rho$.

It essentially means you're taking one and only one element for each equivalence class.

Usually, a classification problem is stated in terms of the relation of being isomorphic on a class of structures.

The issue with this definition of classification is that no work is required to provide a classification. If you ask me for a classification of finite simple groups, I well tell you: “Choose any transversal of the isomorphism equivalence relation on the class of all finite simples groups. Such a transversal exists by the axiom of choice.” (I ignore size issues.) As John wrote later, we must “describe” a representative of each isomorphism class. But without definition of “describe” we still don’t have a satisfying mathematical definition of “classification”.

John Baez (Nov 24 2024 at 04:57):

This is one reason people like to avoid the axiom of choice and other nonconstructive principles: in constructive mathematics, proving the existence of the classification requires actually providing the classification, not just waving the axiom of choice like a magic wand.

Jean-Baptiste Vienney (Nov 24 2024 at 05:04):

That’s a good point but it does not make me feel like I now understand what is a mathematical definition for “classification”. Because if we say that a classification is a transversal defined using “constructive mathematics” (say that we have a fixed definition of constructive mathematics), then I think the classification of finite simple groups doesn’t satisfy this definition. I guess (I don’t really know but I would be surprised if it was not true) that the axiom of choice must be used many times in today’s proof.

Jean-Baptiste Vienney (Nov 24 2024 at 05:05):

Maybe we could say that “describe an isomorphism class” is something like giving a presentation of a representative.

Jean-Baptiste Vienney (Nov 24 2024 at 05:07):

But again we don’t want to be allowed to used the axiom of choice to produce presentations magically.

John Baez (Nov 24 2024 at 05:11):

I don't know if the axiom of choice was used in the classification of finite simple groups. But that's a hard proof that almost nobody understands. Let's ask the experts here about a simpler case: the classification of finite abelian groups? Can people prove the classification of these constructively?

Or, even easier, the classification of finite-dimensional vector spaces over some particular finite field, like $\mathbb{F}_2$. Here most of the classification amounts to showing any finite-dimensional vector space has a basis and two finite-dimensional vector spaces are isomorphic iff and only if there's a bijection between their bases. Since I'm not very good at constructivism I'm not sure these are true constructively!

Mike Shulman (Nov 24 2024 at 05:30):

Is there a definition of "finite-dimensional" other than "has a finite basis"?

John Baez (Nov 24 2024 at 06:15):

How about "finitely generated"? Can one show constructively that every finitely generated module of a field is free on some finite set?

Mike Shulman (Nov 24 2024 at 06:54):

I doubt it. Take the module freely generated by some finitely indexed set; it should be finitely generated, but it seems unlikely to have a finite basis.

Mike Shulman (Nov 24 2024 at 06:56):

Over a finite field, however, you could ask that the vector space itself be a finite set. I would expect this to be stronger than being finitely generated as a module, and it might be sufficient to show that it has a finite basis. I'm likewise not devoid of hope for the classification of finite abelian groups.

Mike Shulman (Nov 24 2024 at 06:57):

However, all this constructive nonsense doesn't seem likely to make it very convincing to the classical mathematician who just wants to know "what is a classification?"

John Baez (Nov 24 2024 at 06:58):

Yes, it would be very interesting if finitely generated modules over a finite field were easy to classify constructively!

So you're saying that in the case of a general field you can't use some algorithm to take a finite set of generators of some vector space over a field, go through them one at a time, find the first that's linearly dependent on the previous ones, throw it out, go on to the next, etc., and get a basis?

Mike Shulman (Nov 24 2024 at 06:59):

On the subject of general classifications, what if one required the set of representatives to be defined constructively, and perhaps to prove constructively that the representatives are pairwise nonisomorphic, but allow using classical logic to prove that everything else is isomorphic to some representative?

Riley Shahar (Nov 24 2024 at 06:59):

John Baez said:

How about "finitely generated"? Can one show constructively that every finitely generated module of a field is free on some finite set?

Consider any binary sequence $a_n$ and define a sequence of fields $F_n := \{x + ia_ny : x, y\in\mathbb{Q}\}$. (Each $F_n$ is a field under the restriction of the field structure on $\mathbb{C}$.) Let $F = \bigcup_n F_n$. Then $F$ is a field, and $\mathbb{Q}(i)$ is an $F$-module under translation. Moreover, $\mathbb{Q}(i)$ is generated by $1$ and $i$ as an $F$-module, but if we could exhibit a basis then we could decide whether any of the $a_n$ are $1$.

Riley Shahar (Nov 24 2024 at 07:00):

(I've seen this example somewhere before, I don't remember where though.)

Mike Shulman (Nov 24 2024 at 07:00):

John Baez said:

So you're saying that in the case of a general field you can't use some algorithm to take a finite set of generators of some vector space over a field, go through them one at a time, find the first that's linearly dependent on the previous ones, throw it out, go on to the next, etc., and get a basis?

Well, I'm not an expert in constructive algebra, but it sets off warning bells in my head. For instance, this algorithm seems like it probably relies on excluded middle to test whether each generator is linearly dependent on the previous ones or not.

Mike Shulman (Nov 24 2024 at 07:01):

For a finite field, you can iterate through all the finitely many linear combinations of any finite set of vectors, but if you want to test whether each of them is zero, you need your vector space to have decidable equality.

John Baez (Nov 24 2024 at 07:02):

By the way, I forgot that even within classical mathematics there's a very nice theory of which classifications can be 'Borel reduced' to which other classifications. This allows us to make precise the idea that some classifications are harder than others:

• John Baez, Wild knots are wildly hard to classify.

'Wild knots' are circles topologically but not necessarily smoothly embedded in $\mathbb{R}^3$. Vadim Kulikov showed that equivalence of countable models of any first-order theory with countably many symbols can be Borel reduced to equivalence of (possibly wild) knots.

This is a precise way of saying that it's hopeless to classify wild knots!

This is just one result in a rather large and fascinating subject.

Mike Shulman (Nov 24 2024 at 07:04):

@Riley Shahar Nice example! That shows that if you allow weird fields, all bets are off. But I bet things can also go wrong even if your field is nice and finite, unless you require the vector spaces to also be finite.

Mike Shulman (Nov 24 2024 at 07:31):

(Do you need the sequence $a_n$ to be increasing, so that the sequence $F_n$ is increasing, so that $F=\bigcup_n F_n$ is a field?)

Riley Shahar (Nov 24 2024 at 07:36):

Mike Shulman said:

(Do you need the sequence $a_n$ to be increasing, so that the sequence $F_n$ is increasing, so that $F=\bigcup_n F_n$ is a field?)

I think it is a field regardless, under the normal operations on $\mathbb{C}$. For instance, take $(x + ia_ny)(x' + ia_my') = xx' - a_na_myy' + i(xa_my' + x'a_ny)$, and since for any $n$ we can decide whether $a_n = 0$ or $a_n = 1$, this decidably defines an element of one of $F_n$ or $F_m$, hence of the union.

David Michael Roberts (Nov 24 2024 at 07:43):

@John Baez even in classical mathematics, but without AC, you have to be careful about the distinction between 'finite' and 'Dedekind finite', and the latter can be very large, eg "it is also consistent with ZF that there is a Dedekind finite, infinite set of reals"

Mike Shulman (Nov 24 2024 at 07:47):

@Riley Shahar Ah, okay, I guess that works.

Riley Shahar (Nov 24 2024 at 07:49):

Mike Shulman said:

Riley Shahar Ah, okay, I guess that works.

Yeah, it's a little trippy, but I think it's fine?

Mike Shulman (Nov 24 2024 at 07:49):

@David Michael Roberts Does that ever actually bite people, though? How often do Dedekind-finite things that aren't finite crop up in practice? The problems with constructive finiteness are relevant because things you expect classically to stay finite, like taking a subset or a quotient of a finite set, don't stay (B-)finite constructively, so subfinite and finitely-indexed sets appear naturally even if you started with finite sets. But I don't know whether something similar happens with Dedekind-finiteness in mathematics with LEM but not AC.

Riley Shahar (Nov 24 2024 at 07:50):

Mike Shulman said:

Riley Shahar Nice example! That shows that if you allow weird fields, all bets are off. But I bet things can also go wrong even if your field is nice and finite, unless you require the vector spaces to also be finite.

Just to complete the example, I think we can use a similar trick for any $\mathbb{k}$. Let $a_n$ be a binary sequence and consider the sequence of $\mathbb{k}$-modules $M_n := \{a_nx : x\in \mathbb{k}\}$. Then $\bigcup_n M_n$ is a $\mathbb{k}$-module via the same trick we used to make $F$ a field.

Let $M := \mathbb{k}/(\bigcup_n M_n)$. Then $M$ is generated by $1 + \bigcup_n M_n$, but if we have a basis then we can decide whether $M = \mathbb{k}$ or $M = *$, so we can decide whether any of the $a_n$ are $1$.

David Michael Roberts (Nov 24 2024 at 09:20):

Mike Shulman said:

The problems with constructive finiteness are relevant because things you expect classically to stay finite, like taking a subset or a quotient of a finite set, don't stay (B-)finite constructively, so subfinite and finitely-indexed sets appear naturally even if you started with finite sets. But I don't know whether something similar happens with Dedekind-finiteness in mathematics with LEM but not AC.

Dedekind finiteness seemingly isn't inherited by quotients (there is an example in a boolean topos (in fact of the form GSet for appropriate G) of a Dedekind finite object with a Dedekind infinite quotient https://doi.org/10.1016/0022-4049(87)90130-7. (It seems in boolean toposes subobjects, which are complemented, of Dedekind finite objects are Dedekind finite)

Also, if there is an infinite Dedekind finite set, then there is a Dedekind finite set whose powerset is Dedekind infinite https://math.stackexchange.com/a/268751/3835

Proofs of properties/structures that rely on induction on the naturals would fail for merely Dedekind finite sets/structures, no doubt. Especially as such sets cannot be well-ordered, so even transfinite induction doesn't help, at least naively.

I don't know of any specific examples to the motivating question, though!

David Michael Roberts (Nov 24 2024 at 09:30):

Actually that first link has an example of a Dedekind finite object in a topos GSet (the same as the other example referenced) with Dedekind infinite power object. Note that the G is in fact a countable group, the free group on a countably infinite set of generators, and the underlying set of the example is the integers. So nothing weird or relying on external independence choices, and the action is very concrete, in terms of generators.

Peva Blanchard (Nov 24 2024 at 10:05):

It seems that classification also depends one the chosen equivalence relation.
Is there a non-trivial example where a hard classification problem is made easier by relaxing the equivalence relation?
(a trivial example is, e.g., the relation with a unique equivalence class)

Jean-Baptiste Vienney (Nov 24 2024 at 13:42):

As far as I understand, the classification of topological spaces is made easier by relaxing the equivalence relation on topological spaces from “homeomorphism” to “homotopy equivalence”.

Jean-Baptiste Vienney (Nov 24 2024 at 13:44):

Mike Shulman said:

On the subject of general classifications, what if one required the set of representatives to be defined constructively, and perhaps to prove constructively that the representatives are pairwise nonisomorphic, but allow using classical logic to prove that everything else is isomorphic to some representative?

I think this is the kind of definition we want!

Andrius Kulikauskas (Nov 24 2024 at 15:26):

@John Baez Thank you for the fascinating classification of free varieties! These are the types of deep examples that I am always looking for. I have included it here in my list of examples of the three minds - answering, questioning, investigating. In my understanding, affine spaces (or vector spaces) are what model our lowest level of sentience, the answering mind (like large language models), the first distillation of reality. A distillation of this distillation (by taking the field to have just one element) yields sets, which model our next level of sentience, the questioning mind (weaving a language of concepts and words). A distillation of the distillation of the distillation (equating all elements of a set) models the final level of sentience, the investigating mind, which matches the answering mind and the questioning mind, and establishes unity.

Andrius Kulikauskas (Nov 24 2024 at 15:31):

The distinction between unpointed and pointed makes this that much more rich in structure. But here the mathematical notion of "pointed" seems inappropriate. In the case of the singletons, the pointed case would be the one without the empty set. But aesthetically speaking, or metaphysically speaking, it seems clear that the plain case is the one without the empty set, where we only have singletons. Including the empty set is like choosing a singleton and removing its element (rather than specifying it). The picture is much more uniform, and I think, satisfying then. Mathematical definitions arise bottom-up and so can miss the point in the big picture. Insisting that "pointed" is the right concept here I think would be a case of being trapped in the mathematics of one's time. All of this to say that I think the truly deep results in mathematics suggest that there is a deeper language than mathematics which is at play. Mathematics is explicitly axiomatic. Whereas mathematical thinking is prior to axiomatization.

Andrius Kulikauskas (Nov 24 2024 at 15:42):

The structure 3+3+1+1 is one that I have observed repeatedly. Recently, I realized that it can be understood as follows. The answering mind models the three minds (say, here, as affine space, set, singleton). The questioning mind likewise models the three minds, here choosing a reference point (an origin, a chosen element, a universal element). (The empty set can be thought of as the one that has a universal element which belongs to all sets.) The investigating mind does not model the three minds but instead models the relationship between the questioning mind and the answering mind, and also models itself, above the fray. An example of this is the 6 needs (of Maslow's hierarchy) and 8 ways of addressing them. I add a diagram and here is my video.
11-ThreeMindsNeeds.png

Andrius Kulikauskas (Nov 24 2024 at 15:44):

The ways of addressing needs happen to be pre-mathematical in spirit, notably, the unpointed cases, from the point of view of the answering mind:

• cling to what you have; get more than what you need; avoid extremes

And the pointed cases, with a reference point, from the point of view of the questioning mind:

• choose the good over the bad; choose the better over the worse; choose the best over the rest

Mike Shulman (Nov 24 2024 at 16:06):

@David Michael Roberts What I meant was, does it ever happen that you start from finite things and end up with only Dedekind-finite things, the same way you can start from finite things and end up with only subfinite or finitely-indexed things? If not, then Dedekind-finite infinite sets seem more like a curiosity, since we can just work with actually-finite things all the time.

John Baez (Nov 24 2024 at 16:36):

Jean-Baptiste Vienney said:

As far as I understand, the classification of topological spaces is made easier by relaxing the equivalence relation on topological spaces from “homeomorphism” to “homotopy equivalence”.

Usually people relax the equivalence relation to weak homotopy equivalence: otherwise the vast array of nasty topological spaces is much too difficult. For example, the rationals with their subspace topology and the rationals with their discrete topology are not homotopy equivalent, but they are weakly homotopy equivalent.

An equivalent approach is to work with homotopy equivalence, but only study 'nice' topological spaces made by gluing balls together, called [[CW complexes]].

Jean-Baptiste Vienney (Nov 24 2024 at 16:39):

Are homotopy equivalence and weak homotopy equivalence the same for CW complexes?

Jean-Baptiste Vienney (Nov 24 2024 at 16:40):

I'm wondering what you mean by "equivalent approach"

Jean-Baptiste Vienney (Nov 24 2024 at 16:41):

Maybe you just mean that it's another way to make "equivalence" of topological spaces easier to study.

Jean-Baptiste Vienney (Nov 24 2024 at 16:43):

But anyway, I should learn about weak homotopy equivalences. The prof didn't talk about this in my algebraic topology class. Maybe because we didn't study higher homotopy groups.

Mike Shulman (Nov 24 2024 at 16:45):

Yes, two CW complexes are homotopy equivalent if and only if they're weak homotopy equivalent, and every space is weak homotopy equivalent to a CW complex. Thus, the weak-homotopy-equivalence classes of topological spaces are the same as the homotopy-equivalence classes of CW complexes.

John Baez (Nov 24 2024 at 21:03):

Jean-Baptiste Vienney said:

Are homotopy equivalence and weak homotopy equivalence the same for CW complexes?

Yes, that's an important theorem probably proved by Whitehead.

Ivan Di Liberti (Nov 24 2024 at 21:17):

John Baez said:

Here's a very nice classification problem in category theory: classify all Lawvere theories for which all finitely generated models are free. There turn out to be 6 kinds. Instead of describing the Lawvere theories syntactically, I'll just say what their models are:

• sets
• pointed sets
• sets where all elements are equal
• pointed sets where all elements are equal
• affine spaces over a chosen division ring
• pointed affine spaces over a chosen division ring

The last one is usually described as "modules over a chosen division ring", and if the division ring is commutative they're usually called "vector spaces over a chosen field". But I wanted to emphasize the parallels.

This classification theorem is apparently not at all trivial.

John Baez said:

Here's a very nice classification problem in category theory: classify all Lawvere theories for which all finitely generated models are free. There turn out to be 6 kinds. Instead of describing the Lawvere theories syntactically, I'll just say what their models are:

• sets
• pointed sets
• sets where all elements are equal
• pointed sets where all elements are equal
• affine spaces over a chosen division ring
• pointed affine spaces over a chosen division ring

The last one is usually described as "modules over a chosen division ring", and if the division ring is commutative they're usually called "vector spaces over a chosen field". But I wanted to emphasize the parallels.

This classification theorem is apparently not at all trivial.

https://www.youtube.com/watch?v=9cGgOmnqzvE&t

David Michael Roberts (Nov 25 2024 at 02:03):

@Mike Shulman oh, I see. I suspect that what you are thinking of doesn't happen, in that case.

Sam Staton (Nov 25 2024 at 06:45):

About open questions. The TLCA open problems are a nice old list, and at least problem number #15 (possibly others) relates to category theory.

given any bicartesian-closed category $\mathcal{C}$ that is not a preorder, the equational theory induced by all interpretations [of the typed lambda calculus with finite sums and products] in $\mathcal{C}$ is exactly beta/eta equality.

http://tlca.di.unito.it/opltlca/

As far as I know, #15 is still open.

David Michael Roberts (Nov 25 2024 at 07:04):

This is not an existing problem, but one that I think would be somewhat interesting: Bourbaki had a definition of 'canonical mapping' for a species of structure (written in a language, with a signature etc or approximately this type of setting). The specifics aren't important here, but it is roughly that the mapping is something close to 'core natural' plus that its graph (I guess as a subset of the cartesian product of 'underlying sets') is definable in the given language using at most the constants in that language (as far as I can tell).

If we considered say, as a warm-up, a Lawvere theory $T$ and the resulting concrete category $\mathrm{Mod}_T$ of models in $\mathbf{Set}$ - can the Bourbaki-canonical morphisms be described using just categorical language?

More generally, given any a general algebraic theory, essentially algebraic theory, generalised algebraic theory ,.... can one isolate and describe the Bourbaki-canonical morphisms? Do these categorical descriptions this recover the examples of morphisms that Bourbaki called canonical, for example the unique (compatible) isomorphism between any two limits/colimits, or more generally the morphism induced by the universal property of such, or the morphisms in the (co)limit (co)cone, or the coherence isomorphisms for things like tensor products?

David Michael Roberts (Nov 25 2024 at 08:14):

Personally I feel Jim Dolan's definition of 'canonical' isn't quite strong enough, even though I believe it's one of the conditions in Bourbaki. BTW, you can only get Bourbaki's definition from an obscure source, I had to get a scan from France.

John Baez (Nov 25 2024 at 08:24):

Strong enough for what? He wasn't trying to match Bourbaki's. definition.

Anyway, your problem is interesting. Say we fix a Lawvere theory $T$ and let $F, G : T \to \mathsf{Set}$ be two models (functors that preserve finite products). We can study the natural transformations $\alpha : F \Rightarrow G$, the transformations $\alpha: F \Rightarrow G$ that are natural with respect to isomorphisms in $T$ (which you're calling core natural), and those that are definable using only 'constants' in $T$. Can we look at all 3 of these in some concrete example, to see how they differ? (I don't understand the third one, so I may be mis-stating it.)

David Michael Roberts (Nov 25 2024 at 08:27):

I just mean that "core natural" on its own isn't strong enough, since any morphism can be put in a context where this is true. It has to be defined to be core natural relative to ... something, eg data about the signature of a theory etc.

David Michael Roberts (Nov 25 2024 at 08:35):

I'm not sure that your setup with natural transformations is the right one, here. Bourbaki asks for 'transportability' of a term along mappings. François Dorais gives a better explanation here https://mathoverflow.net/a/20159/4177

So it's the graph of the function that needs to be definable in terms of constants of the theory, and transportable along isomorphisms (presumably of its domain and codomain, I hope not of their cartesian product)

Joe Moeller (Nov 25 2024 at 21:15):

It would be cool at the main conferences (I'll say ACT specifically since it's closest to my heart) to have an open problem generating and listing/cataloging session. One or three people could volunteer to run the session, writing suggestions on the board and lumping them together on topic, hopefully somehow reflecting the popular topics of the presentations (e.g. categorical probability, quantum computation, networks, etc). The crowd could shoot down ones that are either known or somehow invalid. Then one of the volunteers writes it up and it gets included in the proceedings. The organizers could also encourage speakers to contribute an open problem in their presentation.

David Michael Roberts (Nov 25 2024 at 21:31):

I was reading about algebraic topology in the 30s, and this happened a bit. And there was even a conference that generated two independent open problems lists.

Joe Moeller (Nov 25 2024 at 21:45):

Could it create a problem of the next year having multiple people submit solutions to the same problem? I could imagine ways of avoiding this as a problem. I'm reminded of the story of the HOMFLY polynomial, where multiple people submitted basically the same paper to the same journal, and the editor basically combined them (HOMFLY is the list of authors: Hoste-Ocneanu-Millett-Freyd-Lickorish-Yetter). This story is a triumph of journal editing, but I wouldn't want to indirectly recreate it.

John Baez (Nov 25 2024 at 22:33):

David Michael Roberts said:

I just mean that "core natural" on its own isn't strong enough, since any morphism can be put in a context where this is true. It has to be defined to be core natural relative to ... something, eg data about the signature of a theory etc.

Okay. But James only talks about canonicalness, not of a single morphism, but a transformation (not necessarily natural) between two functors. And this transformation brings its own context, namely the context where we have two categories and two functors from the first to the second.

John Baez (Nov 25 2024 at 22:38):

For example taking the center of a group is not a functor from the category of groups to itself, so it's "not functorial", but it's "core functorial" since it defines a functor $Z$ from the groupoid of groups to itself. And the inclusion of the center in the original group is core natural.

John Baez (Nov 25 2024 at 22:42):

So, we can say the inclusion of the center of a group in that group is "canonical", but that only makes sense in a context, e.g. the context above where we have two functors from the groupoid of groups to itself, namely $Z$ and $1$.

John Baez (Nov 25 2024 at 22:43):

But it's fun to think about "canonical" things you can do with Lawvere theories, too.

David Michael Roberts (Nov 25 2024 at 23:10):

OK, so if he only wants to talk about 'canonical transformations' then I'm cool with that. This doesn't capture what I'm hoping to, but it's still a useful comparison point.

Ryota Kuroki (Nov 29 2024 at 01:38):

It may be too late now, but I will make some remarks on the classification of finitely generated modules in constructive algebra.

• Finitely presented modules over a PID $A$ are well behaved. Any such module is isomorphic to a module of the form $\bigoplus_{i=1}^p A / \langle a_i \rangle \oplus A^r$ ($a_1,\ldots,a_p$ are regular and $a_i \mid a_{i+1}$), and there is a general uniqueness result (Proposition IV-7.3 and Theorem IV-5.1 of the book of Lombardi and Quitté (https://arxiv.org/abs/1605.04832)).
• Let $k$ be a nontrivial discrete field. "For every $k$-submodule $M \subseteq k$, $k/M$ is free" implies LEM. Proof: Let $P$ be a proposition and $M:=\{x \in k : (x=0) \lor P\}$. Then there is a set $X$ such that $k/M \cong \bigoplus_{x \in X} k$. We assume $\neg\neg P$ and prove $P$. First, $X$ cannot be inhabited, because it implies $\exists e\in \bigoplus_{x \in X} k.\ e \neq 0$, $1 \notin M$, and $\neg P$. So $X=\emptyset$, $1 \in M$, and $P$ holds.

Emily (she/her) (Nov 30 2024 at 18:58):

I'm trying to make something like this on Clowder, here.

Emily (she/her) (Nov 30 2024 at 18:58):

So far there isn't really much listed there, just mostly questions that came up while writing the current material.

Emily (she/her) (Nov 30 2024 at 18:58):

However, I do plan to make the list into something more comprehensive in the future (doing things like e.g. going over all category-theory questions on MO), so I'd definitely also love to hear about more open problems in CT

John Baez (Nov 30 2024 at 19:00):

I listed a few open problems here on pages 5, 15 and 20. The first is called a "problem" because it requires finding a good definition, while the other two are called "conjectures".

Emily (she/her) (Nov 30 2024 at 20:03):

Thanks, John! I've saved the slides here and will add the problems listed there on Clowder once I get to the chapters on semi/ring categories :)

Emily (she/her) (Nov 30 2024 at 20:03):

Incidentally, one of the things I want to flesh out then is on defining semiring categories as monoids in a monoidal bicategory, which would possibly give a solution to the problem on page 5

Emily (she/her) (Nov 30 2024 at 20:03):

I remember I got a definition which made 19 out of the 22 coherences hold, but never got to write it down in an organized way, nor try to find an alternative definition which got all 22 coherences.

John Baez (Nov 30 2024 at 20:27):

@Emily (she/her) - I'd been thinking of sharing a list of open problems on the n-Category Cafe, but now I may link to yours.

Riley Shahar (Nov 30 2024 at 20:38):

I don't think I've seen this mentioned here: the nlab has a list of [[open problems in homotopy type theory]]; it seems like a natural place to host an analogous list for (higher) category theory.

Emily (she/her) (Nov 30 2024 at 21:42):

John Baez said:

Emily (she/her) - I'd been thinking of sharing a list of open problems on the n-Category Cafe, but now I may link to yours.

I'm happy you find it useful enough to link to! Hopefully I'll be able to make it into something more comprehensive sooner rather than later :)

Emily (she/her) (Nov 30 2024 at 21:42):

Having a post over the nCatCafé would also likely prompt more people to list more problems, so that would be really great I think!

Emily (she/her) (Nov 30 2024 at 21:42):

(And, for what it's worth, I'd really love to have a look at the list you had in mind :)

John Baez (Nov 30 2024 at 21:57):

It would be just the list we assembled here, unless I can think of anything else - so I'm glad you've (apparently) already copied down this list. My post at the n-Category Cafe will also be a request for people to contribute more problems.

John Baez (Nov 30 2024 at 21:57):

I remember I got a definition which made 19 out of the 22 coherences hold, but never got to write it down in an organized way, nor try to find an alternative definition which got all 22 coherences.

Ha! I remember someone saying that on MathOverflow, but I didn't remember it was you.

John Baez (Nov 30 2024 at 22:02):

Emily (she/her) said:

Thanks, John! I've saved the slides here and will add the problems listed there on Clowder once I get to the chapters on semi/ring categories :)

Great. I'll wait until you tell me you're approximately done with your list, and then I'll announce it on the n-Cafe.

Incidentally, one of the things I want to flesh out then is on defining semiring categories as monoids in a monoidal bicategory, which would possibly give a solution to the problem on page 5

I remember I got a definition which made 19 out of the 22 coherences hold, but never got to write it down in an organized way, nor try to find an alternative definition which got all 22 coherences.

I find it incredible that one could get 19 out of 22 coherence laws but not all. Defining a [[rig category]] to be a [[pseudomonoid]] in the (symmetric) monoidal bicategory of symmetric monoidal categories seems like a great idea: I think that's essentially what you're suggesting here.