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Stream: learning: questions

Topic: ✔ 2 limits in 2-categories of diagrams

Nico Beck (Jun 09 2023 at 08:58):

does somebody know which strict (i.e. Cat-enriched) weighted (co)limits the categories Ps(K,Cat) and Hom(K,Cat) repectively have? here K is a strict 2 category, or even a 1 category if it helps. I know that there is a 2 monad T on $\prod_{ob K} \operatorname{Cat}$ such that the two 2 categories are TAlg and PsTAlg repsectively, whih implies that they both have all (PIE)-weighted limits. I have also seen that Hom(K,Cat) has all (flexible)-weighted limits. What about strict colimits though?

John Baez (Jun 09 2023 at 14:27):

I don't actually know the following, but I hope it's true:

Conjecture. If V is a Benabou cosmos (a complete and cocomplete closed symmetric monoidal category) and K is a V-category, then Hom(K,V) has all small weighted limits and colimits.

John Baez (Jun 09 2023 at 14:27):

If it's true you could take V = Cat and the answer to your last question would be "yes".

John Baez (Jun 09 2023 at 14:28):

Someone should have studied this since Hom(K,V) is what I'd call an "enriched presheaf category" and they're pretty important.

Mike Shulman (Jun 09 2023 at 14:59):

I think by Hom(K,Cat) Nico means the 2-category of pseudofunctors and pseudonatural transformations. That may have strict coproducts, but I doubt it has any other strict limits.

Mike Shulman (Jun 09 2023 at 15:00):

(By the way, Nico, you need two dollar-signs around all math here.)

John Baez (Jun 09 2023 at 15:02):

Okay, I guess I don't know what Nico meant by "the categories Ps(K,Cat) and Hom(K,Cat), respectively".

Joe Moeller (Jun 09 2023 at 15:03):

Wait, if that's Hom(K,Cat), then what is Ps(K,Cat)?

Mike Shulman (Jun 09 2023 at 15:04):

Strict 2-functors and pseudonatural transformations. I didn't immediately know what he meant by either of those notations, but then when he said "there is a 2 monad T on $\prod_{{\rm ob} K} \rm Cat$ such that the two 2 categories are TAlg and PsTAlg repsectively" I reached this guess.

Mike Shulman (Jun 09 2023 at 15:06):

Because I know there is such a 2-monad such that $T {\rm Alg}_s$ (the 2-category of strict $T$-algebras and strict $T$-morphisms) consists of strict 2-functors and strict 2-natural transformations, $T{\rm Alg}$ (the 2-category of strict $T$-algebras and pseudo $T$-morphisms) consists of strict 2-functors and pseudonatural transformations, and ${\rm Ps}T{\rm Alg}$ (the 2-category of pseudo $T$-algebras and pseudo $T$-morphisms) consists of pseudofunctors and pseudonatural transformations.

Mike Shulman (Jun 09 2023 at 15:06):

And those notations for those 2-categories associated to a 2-monad are fairly standard in 2-monad theory.

Nico Beck (Jun 09 2023 at 23:16):

Mike Shulman said:

I think by Hom(K,Cat) Nico means the 2-category of pseudofunctors and pseudonatural transformations. That may have strict coproducts, but I doubt it has any other strict limits.

yes, exactly. by $Ps(K,\operatorname{Cat})$ I mean the 2-category of strict 2-functors, pseudo natural transformations and modifications, and by $Hom(K,\operatorname{Cat})$ I mean the 2-category of pseudo functors, pseudo trnadformations and modifications.

Nico Beck (Jun 09 2023 at 23:25):

Mike Shulman said:

I think by Hom(K,Cat) Nico means the 2-category of pseudofunctors and pseudonatural transformations. That may have strict coproducts, but I doubt it has any other strict limits.

The 2–category $Ps(K,Cat)$ has all (PIE)-weighted strict limits, and $Hom(K,Cat)$ has all (flexible)-weighted strict limits. this is mentioned in remark 7.4 of "flexible limits for 2-categories". in general 2 categories of the shape TAlg for a 2-monad do not have many strict colimits, but I was under the impresion that $Hom(K,Cat)$ is also the 2 category of pseudo coalgebras for some 2 comonad. but I don't remember where I read this. So I thought maybe Hom(K,Cat) has some strict colimits?

Nico Beck (Jun 09 2023 at 23:31):

Mike Shulman said:

And those notations for those 2-categories associated to a 2-monad are fairly standard in 2-monad theory.

yes. I am following the notation conventions of the papers on 2-limits and 2-monads by Kelly, Power, Street, Lack, et al. Ps(K,Cat) is an in between thing, where the 0-cells are the strict ones, but the 1-cells are pseudo

Mike Shulman (Jun 10 2023 at 01:52):

However, your notations Ps(K,Cat) and Hom(K,Cat) are not very standard, so it's probably a good idea to tell your readers what you mean by them.

Mike Shulman (Jun 10 2023 at 01:55):

And I wouldn't recommend using "Hom(K,Cat)" for the 2-category of pseudofunctors in any case; you may find this in some older papers that use Benabou's term "homomorphism (of bicategories)" for pseudofunctor, but that's pretty well deprecated by now.

Mike Shulman (Jun 10 2023 at 01:59):

But you're right that these are also categories of coalgebras for a 2-comonad; I should have remembered that. I guess that does mean that Ps(K,Cat), at least, has PIE-colimits, since PIE-limits lift to TAlg for any 2-monad T. I'm less sure about Hom(K,Cat) since I don't know a general theorem about limits lifting to categories of pseudoalgebras for a 2-monad, but probably it's true?

Mike Shulman (Jun 10 2023 at 02:08):

By the way, I think it's good form to mention and link when you cross-post a question on multiple forums (https://mathoverflow.net/q/448467/49).

Nico Beck (Jun 10 2023 at 05:29):

Yes, I am not sure about the flexible colimits. (And I am also not sure about the PIE-colimits because I do not understand most of it well enough at the moment). The result about PIE-limits looks like it works for any 2 monad and thus should also work for comonads and colimits, but the flexible limits seem to depend on replacing the 2 monad T by its flexible replacement T' in the category of finitary monads, and I am not sure that that also works for the comonads, because I do not understand all the finiteness conditions well enough. I thought maybe someone knows which strict colimits exist and how they are formed. maybe there is even a simpler argument for categories of diagrams, and maybe one can show that they are formed pointwise and are created by the 2functor into $\prod_{obK} \operatorname{Cat}$. That would be really cool. I remember that I have read the statement that at least the bi(co)limits are pointwise formed in the PhD thesis "Categorical Properties od Differentiable and Topological Stacks", but it did not contain a proof. An equivalent question would be what kind of strict 2-(co)limits Fib(B) has, where B is some base category. I believe this notation is standard.

Nico Beck (Jun 10 2023 at 05:34):

Mike Shulman said:

By the way, I think it's good form to mention and link when you cross-post a question on multiple forums (https://mathoverflow.net/q/448467/49).

I am sry about the cross posting! I normally don't do that, but I learned about the zulip server here only yesterday after I already already posted my question on mathoverflow. Since I spend a lot of time last week trying to understand the 2-colimits, I thought I'll try my luck again :)

Mike Shulman (Jun 10 2023 at 05:36):

I don't think cross-posting is wrong, it's just nice to let people know about it. (At least in one direction, e.g. mentioning the MO question here.)

Mike Shulman (Jun 10 2023 at 05:39):

Nico Beck said:

The result about PIE-limits looks like it works for any 2 monad and thus should also work for comonads and colimits

Yes, agreed.

but the flexible limits seem to depend on replacing the 2 monad T by its flexible replacement T' in the category of finitary monads, and I am not sure that that also works for the comonads

Yeah, this doesn't work as stated for comonads; I don't think there is an analogue of T' for comonads. But it might be possible to prove "by hand" that PIE or flexible limits lift to categories of pseudoalgebras for a monad, and then dualize that.

An equivalent question would be what kind of strict 2-(co)limits Fib(B) has, where B is some base category.

I don't think that is equivalent: ${\rm Fib}(B)$ is equivalent to your ${\rm Hom}(B^{\rm op},\rm Cat)$ as a bicategory, but not as a 2-category, and equivalences of bicategories don't preserve strict 2-(co)limits.

Nico Beck (Jun 10 2023 at 05:51):

Mike Shulman said:

Nico Beck said:

The result about PIE-limits looks like it works for any 2 monad and thus should also work for comonads and colimits

Yes, agreed.

but the flexible limits seem to depend on replacing the 2 monad T by its flexible replacement T' in the category of finitary monads, and I am not sure that that also works for the comonads

Yeah, this doesn't work as stated for comonads; I don't think there is an analogue of T' for comonads. But it might be possible to prove "by hand" that PIE or flexible limits lift to categories of pseudoalgebras for a monad, and then dualize that.

An equivalent question would be what kind of strict 2-(co)limits Fib(B) has, where B is some base category.

I don't think that is equivalent: ${\rm Fib}(B)$ is equivalent to your ${\rm Hom}(B^{\rm op},\rm Cat)$ as a bicategory, but not as a 2-category, and equivalences of bicategories don't preserve strict 2-(co)limits.

Theorem 10.6.16 in "2-dimensional categories" states that the Grothedieck construction is a Cat-enriched equivalence $Hom(B^{op},\rm Cat)\to Fib(B)$

Mike Shulman (Jun 10 2023 at 15:32):

Hmm... I guess you're right. I must be thinking of something else.

I'm not doing so well today! (-:

Nico Beck (Jun 14 2023 at 11:54):

No problem, you helped me anyway, thx! I settled on being satisfied with (PIE)-weighted colimits :)

Notification Bot (Jun 14 2023 at 11:58):

Nico Beck has marked this topic as resolved.