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Stream: community: general

Topic: How many structures?

Avi Levy (Sep 16 2020 at 04:12):

John Baez said:

Nathanael Arkor said:

A category can have multiple nonequivalent closed structures.

Yes. Here's a nice result: there are exactly two symmetric monoidal closed structures on the category Cat, up to equivalence. One is the usual cartesian closed structure; the other comes from the funny tensor product of categories.

My gut reaction to this is surprise that there is no way of somehow reversing arrows to obtain a different cartesian closed structure. The "exactly two" structures - is this up to some sort of equivalence, or "on the nose"?

John Baez (Sep 16 2020 at 05:38):

I think my sentence answered that: I said "up to equivalence".

John Baez (Sep 16 2020 at 05:43):

You can always take any monoidal structure and change the tensor product, replacing the tensor product $x \otimes x'$ of each pair of objects by a new isomorphic object, call it $x \boxtimes x'$. If we choose a specific isomorphism $\phi_{x,x'} : x \otimes x' \to x \boxtimes x'$ for each pair $x,x'$, we obtain an equivalence of monoidal categories.

Equivalence of monoidal categories is a standard notion; see e.g. Definition 12 here.

Jules Hedges (Sep 16 2020 at 08:51):

I always found it a bit mind blowing that there's known to be exactly 2. I never tried to find the proof but I really can't imagine what it would look like. I can't think of a any other "classification theorem" like that in category theory

Jules Hedges (Sep 16 2020 at 08:52):

This reminds me of when a bunch of people in Oxford tried to prove that there is only one compact closed structure on finite dimensional vector spaces. If I remember correctly I think they succeeded, but I think the proof was really nasty

John van de Wetering (Sep 16 2020 at 09:44):

Do you mean this paper: https://arxiv.org/abs/1803.00708v1

John van de Wetering (Sep 16 2020 at 10:01):

So, related to the question of having a database of categories: how do people feel about having a 'database' of (A)CT papers?

John van de Wetering (Sep 16 2020 at 10:01):

By that I mean something similar to our list of publications regarding the ZX-calculus: http://zxcalculus.com/publications.html

John van de Wetering (Sep 16 2020 at 10:03):

If people would be interested in making something like that for ACT, I'm happy to talk you through the source code we used to generate that (which is available at https://github.com/zx-outreach/website). It basically boils down to having a well-prepared bib file

Reid Barton (Sep 16 2020 at 10:20):

Here's a similar sort of "classification theorem" with a bigger number: The nine model category structures on the category of sets

Alexander Campbell (Sep 16 2020 at 10:33):

Jules Hedges said:

I always found it a bit mind blowing that there's known to be exactly 2. I never tried to find the proof but I really can't imagine what it would look like. I can't think of a any other "classification theorem" like that in category theory

The proof is really quite nice. The original paper of Kelly et al. is rather light on details, but the exposition in §2 of http://tac.mta.ca/tac/volumes/30/11/30-11abs.html is nicely readable.

Alexander Campbell (Sep 16 2020 at 10:35):

It's worth remarking that the cartesian and funny monoidal structures are the only two monoidal closed structures on Cat, i.e. one needn't say "symmetric".

Jules Hedges (Sep 16 2020 at 11:08):

John van de Wetering said:

Do you mean this paper: https://arxiv.org/abs/1803.00708v1

I think so

Peter Arndt (Sep 16 2020 at 15:07):

Reid Barton said:

Here's a similar sort of "classification theorem" with a bigger number: The nine model category structures on the category of sets

Another one for the collection of crazy numbers: There are just three possible lengths of maximal chains of adjoint functors between compactly generated tensor-triangulated categories, namely 3, 5 and infinity. This comes from here: https://arxiv.org/abs/1501.01999

Morgan Rogers (he/him) (Sep 16 2020 at 15:10):

This thread is getting weirder and weirder

Jules Hedges (Sep 16 2020 at 15:37):

Numbers? In my category theory?

Nikolaj Kuntner (Sep 16 2020 at 15:39):

Call it numberiods and be at ease :)

Peter Arndt (Sep 16 2020 at 17:08):

Since we are at it: The only category for which the Yoneda embedding is the rightmost of a string of 5 adjoints is the category of Sets: https://doi.org/10.1090/S0002-9939-1994-1216823-2

Dan Doel (Sep 16 2020 at 17:31):

Yeah, there are some other things like that in the adjoint string article.

John Baez (Sep 16 2020 at 18:28):

Jules Hedges said:

I always found it a bit mind blowing that there's known to be exactly 2. I never tried to find the proof but I really can't imagine what it would look like. I can't think of a any other "classification theorem" like that in category theory

This is a bit similar: what are all the symmetries of $\mathsf{Cat}$? That is, what are all the equivalences $F: \mathsf{Cat} \to \mathsf{Cat}$, up to natural isomorphism?

It turns out there are just two.

It's not so hard for me to believe these results. Everything about categories and functors is built from a few building blocks. You can completely "probe" a category by looking at maps from the walking object and the walking arrow into this category, and you can also probe what a functor or natural transformation does this way. I've thought more about the result I just mentioned than the other one, about symmetric monoidal closed structures.

Morgan Rogers (he/him) (Sep 16 2020 at 18:28):

Peter Arndt said:

Since we are at it: The only category for which the Yoneda embedding is the rightmost of a string of 5 adjoints is the category of Sets: https://doi.org/10.1090/S0002-9939-1994-1216823-2

This seems circular. Without having read the article, if I were to use a different (perhaps Boolean) topos in place of Set when constructing presheaves, wouldn't that give a comparable result?

Dan Doel (Sep 16 2020 at 18:42):

@[Mod] Morgan Rogers I think you can probably get similar results like that. Maybe you need to consider enriched/internal categories.

Nathanael Arkor (Sep 16 2020 at 18:44):

I would have assumed this property was related specifically to the proarrow equipment structure of the Yoneda embedding on Set, rather than the topos structure.

Dan Doel (Sep 16 2020 at 18:44):

E.G. I came across that adjoint string by working out that it seemed to exist for a 'category of types' that is considerably weaker in a lot of ways than Set.

Jules Hedges (Sep 16 2020 at 18:53):

John Baez said:

This is a bit similar: what are all the symmetries of $\mathsf{Cat}$? That is, what are all the equivalences $F: \mathsf{Cat} \to \mathsf{Cat}$, up to natural isomorphism?

It turns out there are just two.

This one at least I find intuitive, I guess they are basically identity and op. (Like how $\mathbb C$ has 2 automorphisms)

Dan Doel (Sep 16 2020 at 18:53):

So perhaps 'if V is self-enriched and B is a V-category and the V-enriched Yoneda embedding for B is on the right of an adjoint 5-tuple, then B is equivalent to V'.

Nathanael Arkor (Sep 16 2020 at 19:03):

Perhaps for any Yoneda structure where $よ_{\mathbf C}$ is the rightmost of an adjoint string of length 5, then $\mathbf C \simeq \widehat{\mathbf 1}$ (where $よ$ is the Yoneda embedding, and $\widehat{(-)}$ is the presheaf construction).

Oscar Cunningham (Sep 16 2020 at 19:27):

Analogously, do we know that there is only one symmetry of $\mathbf{Set}$?

John Baez (Sep 16 2020 at 19:36):

I haven't proved it but I think so.

The groupoid $\mathsf{S}$ of finite sets and bijections is much more interesting as far as symmetries go! There's an equivalence $F: \mathsf{S} \to \mathsf{S}$ that's not naturally isomorphic to the identity, which does nothing to objects and nothing to morphisms except for bijections between sets with 6 elements.

Even cooler, my last sentence would become false if I replaced the number 6 by any other number!!!

• Some thoughts on the number 6.

Even cooler, the groupoid of finite sets and bijections is the free symmetric monoidal category on one object.

So, if you want to scare people, you can say:

The free symmetric monoidal category on one object $x$ has just one nontrivial autoequivalence modulo natural isomorphisms, and this autoequivalence can be chosen to be the identity except on $\mathrm{hom}(x^{\otimes 6}, x^{\otimes 6})$.

Morgan Rogers (he/him) (Sep 16 2020 at 19:36):

Nathanael Arkor said:

Perhaps for any Yoneda structure where $よ_{\mathbf C}$ is the rightmost of an adjoint string of length 5, then $\mathbf C \simeq \widehat{\mathbf 1}$ (where $よ$ is the Yoneda embedding, and $\widehat{(-)}$ is the presheaf construction).

Dan Doel said:

So perhaps 'if V is self-enriched and B is a V-category and the V-enriched Yoneda embedding for B is on the right of an adjoint 5-tuple, then B is equivalent to V'.

Something like these! In any case, the point is that this is saying more about Yoneda embeddings than Set, imo

Dan Doel (Sep 16 2020 at 19:38):

Yeah, I suspect so.

Jens Hemelaer (Sep 16 2020 at 20:20):

Oscar Cunningham said:

Analogously, do we know that there is only one symmetry of $\mathbf{Set}$?

I would think that any symmetry $\sigma$ of $\mathbf{Set}$ extends to a symmetry of $\mathbf{Cat}$, by letting the symmetry act on the set of objects and the set of morphisms. Does this work?

Reid Barton (Sep 16 2020 at 20:27):

Any automorphism of Set has to preserve coproducts and the terminal object; and any object $S$ of Set is the coproduct of $S$ copies of the terminal object.

John Baez (Sep 16 2020 at 20:27):

Jens' argument works! If we extend them this way, nonisomorphic autoequivalences of Set will give nonisomorphic autoequivalences of Cat, since Set sits inside Cat as the discrete categories. So, there can't be more than 2 nonisomorphic autoequivalences of Set. But no autoequivalence of Set gives the nontrivial autoequivalence of Cat, namely "op".

Nathanael Arkor (Sep 16 2020 at 20:30):

Reid Barton said:

Any automorphism of Set has to preserve coproducts and the terminal object; and any object $S$ of Set is the coproduct of $S$ copies of the terminal object.

May similar reasoning can be applied to prove that Cat only has two automorphisms? Cat is a locally presentable category, whose finitely presentable object is $\bullet \to \bullet$, and there are two natural actions on this object: taking the identity, or flipping the direction of the arrow.

Jens Hemelaer (Sep 16 2020 at 20:41):

Reid Barton said:

Any automorphism of Set has to preserve coproducts and the terminal object; and any object $S$ of Set is the coproduct of $S$ copies of the terminal object.

This is much more elegant :smiley:

John Baez (Sep 16 2020 at 20:43):

Nathanael Arkor said:

Presumably the same reasoning can be applied to any locally presentable category, which then leads to the two automorphisms of Cat, given by the two functors on the category $\bullet \to \bullet$.

I'm getting confused. What are the the two functors on this category? (I'm not even sure what "functor on a category" means here!)

Nathanael Arkor (Sep 16 2020 at 20:48):

I've clarified my comment: I was thinking that the automorphisms should be generated by either taking the identity or flipping the direction of the arrow.

John Baez (Sep 16 2020 at 20:54):

Okay... notice that this "walking arrow" category is equivalent to its opposite, so any autoequivalence $F: \mathsf{Cat} \to \mathsf{Cat}$ maps this category to itself (up to natural isomorphism).

John Baez (Sep 16 2020 at 20:56):

There's not really a way to map this category to itself "with the direction of the arrow flipped", as far as I can tell.

(I really do find this confusing, I'm not just testing people.)

Nathanael Arkor (Sep 16 2020 at 21:03):

Perhaps I'm barking up the wrong tree, but if you look at the action of an automorphism F : Cat → Cat on 2 (the walking arrow) we can have F(2) = 2 or F(2) = 2°. Cat is generated under colimits by 2, or equivalently 2°, and it seems to me that a category formed by some colimit of 2 is going to be the dual of a category formed by the analogous colimit of 2°. So one choice of F determines the identity automorphism, and the other choice of F determines the duality involution.

John Baez (Sep 16 2020 at 21:11):

I'm just saying that 2° is isomorphic to 2. There are lots of categories isomorphic to 2, and 2° doesn't play any distinguished role among these, as far as I can tell.

Another way to say it:

If we're studying the group of autoequivalences of Cat mod natural isomorphism, we can replace the category Cat with a skeleton without changing this group. In the skeleton we have 2°=2.

Sam Staton (Sep 16 2020 at 21:52):

Isn't the point that the endomorphism monoid of 2 in Cat has two automorphisms? In other words the one-object full subcategory of Cat has two endo-equivalences? and it must all follow from this? I'm tired, though

Morgan Rogers (he/him) (Sep 17 2020 at 09:28):

John Baez said:

This is a bit similar: what are all the symmetries of $\mathsf{Cat}$? That is, what are all the equivalences $F: \mathsf{Cat} \to \mathsf{Cat}$, up to natural isomorphism?

It turns out there are just two.

Important caveat: these are symmetries as 1-categories, since op reverses natural transformations. The generator argument works, therefore, since any automorphism is determined (up to natural isomorphism) by an automorphism of the generating full subcategory $1 \rightrightarrows 2$, of which there are clearly exactly two.

Jens Hemelaer (Nov 20 2020 at 18:41):

Jules Hedges said:

Somewhere we had a thread about classification theorems in category theory, and this would be a perfect addition there. But I can't find it

As suggested by @Jules Hedges , I'm adding the result by Trnková and Reiterman to this older topic.

A category is called rich if $\mathbf{PSh}(\mathcal{C})$ contains the topos of directed graphs as a full subcategory. What Trnková and Reiterman showed is that a thin category is rich if and only if it contains one of the following 35 thin categories as a full subcategory:
basic-thin.png

See the discussion here.

John Baez (Nov 20 2020 at 20:12):

That's cool!!!