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Stream: learning: questions

Topic: proving some 2-categories are lfp

Matteo Capucci (he/him) (Jan 13 2025 at 15:59):

I'm looking for some guidance/check-my-work on how to go on about proving/disproving the following facts:

$\cal SMC$ , the 2-category of symmetric monoidal categories, braided monoidal functors and monoidal natural transformations is LFP
${\cal L}/X$ is LFP when $\cal L$ is LFP

For (1) I suspect Theorem 6.9 from Bird's thesis might be useful: it states that a monad of rank $\alpha$ on a locally $\alpha$ -presentable (V-)category $\cal K$ yields a L $\alpha$ P category of algebras, and generators are given by free algebras over the generators of $\cal K$ . Now $\cal SMC$ should be such a category: for $V=\bf Cat$ , there is a (finitary afaik) 2-monad whose strict algebras are SMCs.
Also, intuitively $F{\bf 1} \cong \bf S$ (the [[symmetric groupoid]]) and $F{\bf 2}$ (...something else, I don't really care tbh) still pick out objects and maps of an SMC and thus allow to present it like categories are. However I am a bit confused by cardinalities here, since I don't expect every SMC (or every category...) to be a countable directed colimit of objects and maps... or is small directed colimit enough?

Regarding (2)... Bird's thesis linked above proves V-Loc is closed under a bunch of Cat-limits, among which are products and inserters. I wonder if commas can be built out of those, like pullbacks can be built out of products and equalizers. So the comma of $F:C \to D$ and $G:B \to D$ would be constructed as the inserter of $C \times B \rightrightarrows D$ where the top arrow if $F\pi_1$ and the bottom is $G\pi_2$ . This seems to work? And to pick out an object $X \in \cal L$ I would map out of $\bf FinSet$ , with the assignment $A \mapsto A \otimes X = \coprod_{a \in A} X$ , since then ${\cal L}(A \otimes X, L) \cong A \pitchfork {\cal L}(X, L) = {\bf FinSet}(A, {\cal L}(X, L))$ though I don't see why ${\cal L}(X,L)$ should be finite so maybe I'm messing up something (even assuming $X$ is finitely presentable doesn't seem to help..?).

Nathanael Arkor (Jan 13 2025 at 18:28):

I'm on the move, so can't give more details now, but I recommend checking out Bourke's Accessible aspects of 2-category theory, which is about this kind of question.

Kevin Carlson (Jan 13 2025 at 19:21):

You’re talking about braided pseudo monoidal functors, Matteo? Generally you’ll need everything strict to get LFP, because for instance with pseudo, there’s no strict initial object—you get to choose coherences for the pseudo-unique map out of the pseudo-initial guy.

Matteo Capucci (he/him) (Jan 14 2025 at 10:01):

Kevin Carlson said:

You’re talking about braided pseudo monoidal functors, Matteo?

Yes

Matteo Capucci (he/him) (Jan 14 2025 at 10:18):

Kevin Carlson said:

Generally you’ll need everything strict to get LFP, because for instance with pseudo, there’s no strict initial object—you get to choose coherences for the pseudo-unique map out of the pseudo-initial guy.

Ah thanks, I missed that subtlety in Bird's theorem---it applies to strict algebras and strict morphisms, of course.

Matteo Capucci (he/him) (Jan 14 2025 at 10:18):

The reference Nathanael pointed to seems extremely relevant

Matteo Capucci (he/him) (Jan 14 2025 at 10:19):

In particular I don't really need locally finitely presentable, accessible + some specific colimits should be enough for my aims

Kevin Carlson (Jan 14 2025 at 18:46):

Yes, I was quite impressed to learn that these guys are accessible, a dramatic expansion of the field of play of accessibility \ presentability!

Ivan Di Liberti (Jan 15 2025 at 16:55):

Check Examples 6.3.6 and 6.4.6 in my paper with Axel Osmond "Biaccessible and bipresentable 2-categories". That may provide inspiration.