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Stream: deprecated: id my structure

Topic: category of all small diagrams

Brendan Murphy (Feb 15 2023 at 23:09):

Let $\mathsf{Cat}$ be the category of all small categories and $\mathsf{CAT}$ the category of all locally small categories. We have a restricted yoneda embedding $Y : \mathsf{CAT} \to [\mathsf{Cat}^{\mathrm{op}}, \mathsf{CAT}]$. Define $\mathrm{Diag} : \mathsf{CAT} \to \mathsf{CAT}$ by $\mathrm{Diag}(\mathsf{C}) = \int Y(\mathsf{C})$. Explicitly, $\mathrm{Diag}(\mathsf{C})$ has objects pairs $(\mathsf{J}, F)$ where $\mathsf{J}$ is a small category and $F : \mathsf{J} \to \mathsf{C}$, and a morphism $(\mathsf{J}, F) \to (\mathsf{K}, G)$ is a pair $(p, \alpha)$ where $p : \mathsf{J} \to \mathsf{K}$ is a functor and $\alpha : F \to G \circ p$ a natural transformation.

Does anyone recognize this construction? If we look at only discrete diagrams we get the free cocompletion of $\mathsf{C}$. We also have a functor from $\mathrm{Diag}(\mathsf{C})$ to the category $\mathcal{P} \mathsf{C}$ of small presheaves on $\mathsf{C}$ (i.e. the free cocompletion), and I'm wondering if this identifies $\mathcal{P} \mathsf{C}$ with a localization of $\mathrm{Diag}(\mathsf{C})$. Can we say $\mathrm{Diag}(\mathsf{C})$ is the free something on $\mathsf{C}$? Also note that this is the lax slice category $\mathsf{Cat}//\mathsf{C}$, and so is what nlab calls an "F-category".

I think that $\mathrm{Diag}$ is a $(2, 1)$-monad, because we have the constant diagram functor $\mathsf{C} \to \mathrm{Diag}(\mathsf{C})$ and also something that feels like a join $\mu : \mathrm{Diag}(\mathrm{Diag}(\mathsf{C})) \to \mathrm{Diag}(\mathsf{C})$, where the new indexing category of the diagram $\mu(A)$ is the grothendieck construction of $\pi\circ A$, where $\pi : \mathrm{Diag}(\mathsf{C}) \to \mathsf{Cat}$ sends a diagram to its indexing category. If I'm right thinking this is a monad, does it arise from an adjunction?

Ivan Di Liberti (Feb 15 2023 at 23:15):

https://arxiv.org/pdf/2101.04531.pdf

Brendan Murphy (Feb 15 2023 at 23:33):

Sweet, thanks!

Brendan Murphy (Feb 16 2023 at 00:20):

Hm, this is still kind of unsatisfying to me. But some of the thing I want to know (eg what are the algebras) seem to be unknown. I'd like to have a snappy description like "Diag is the free oplax cocompletion" or "Diag is the monad associated to the adjunction ..." but oh well

Brendan Murphy (Feb 16 2023 at 21:28):

I'm trying to think about the algebras over this monad today, and from looking at papers it seems like there's some theory of strictification for this stuff, so eg if you have a strict monad M on Cat (ie a Cat enriched monad), the category of strict M algebras has a model structure that presents the (2, 1)-category of pseudo M-algebras. But the monad I'm considering is weak itself, so I guess the best I could hope for is a strict 2 monad M' whose strict algebras are the pseudoalgebras over my pseudomonad M. Does anyone know a result like this? I've been getting really confused trying to look through the literature by all the various strictness levels

Paolo Perrone (Feb 17 2023 at 10:42):

I'm not entirely sure what you mean exactly by "free oplax cocompletion", but if you mean "adding all lax colimits freely", then Diag is not it. Lax colimits are weighted colimits, and any free construction in that sense would give a lax-idempotent 2-monad, which Diag is not.
If instead you mean something like a "lax version of free cocompletion", that sounds more likely (but I don't know how to make it precise yet).

Evan Patterson (Feb 20 2023 at 04:01):

This paper by Rene Guitart and Luc Van den Bril seems to claim that Diag is a lax cocompletion of some kind. It's written in French, so I can't read it, but the review in English on zbMath says that the 2-category Diag(X) is "a free lax-co-completion of X." It would be great if someone could explain what this means.