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Stream: learning: questions

Topic: Presheaves from functors

Fabrizio Romano Genovese (Mar 04 2024 at 16:36):

I don't know how to formulate this question any better, but given a lax functor between bicategories $F: C \to D$, are there canonical/common/known/studied ways to define a presheaf from $F$?

Fabrizio Romano Genovese (Mar 04 2024 at 16:37):

Ideally, I'd like to have a presheaf that is a sheaf when $F$ is pseudo or strict, but really anything goes at this stage, lol

Xuanrui Qi (Mar 08 2024 at 06:03):

That wouldn't quite make sense because presheaves are (contravariant) functors from $\mathcal{C}$ to $\mathbf{Set}$.

Benjamin Merlin Bumpus (he/him) (Mar 08 2024 at 12:32):

Well, I think the question is assuming whatever suitable notion of op, co etc that might make the above work.

John Baez (Mar 08 2024 at 17:15):

Let's start with something really obvious! Are we talking about presheaves on the bicategory $C$, in which case I need to find out what Fabrizio means by a presheaf on a bicategory... or presheaves on some category?

Here's something we can do. Any pair of objects $x, y \in C$ give a category $\mathrm{hom}(x,y)$. Can we get a presheaf on this category somehow, using $F: C \to D$? Well, $F$ maps $\mathrm{hom}(x,y)$ to the hom-category $\mathrm{hom}(Fx,Fy)$, and any object $f \in \mathrm{hom}(Fx,Fy)$ defines a representable presheaf on that hom-category, which pulls back to give a presheaf on $\mathrm{hom}(x,y)$.

In short, given objects $x, y \in C$ and a morphism $f : Fx \to Fy$ we get a presheaf $P$ on the category $\mathrm{hom}(x,y)$ which on objects has

$P(g) = \mathrm{hom}(Fg, f)$

Mike Stay (Mar 08 2024 at 17:31):

If $D$ is a bicategory of categories equipped with extra structure/stuff, then $C$ might be enriched in $D$. Then you could consider $F$ itself a $D$-presheaf on $C^{\rm op}$.

Mike Stay (Mar 08 2024 at 17:33):

If you have a $\rm Cat$-presheaf $G:D^{\rm op} \to {\rm Cat}$, you can pull it back to $G \circ F^{\rm op}:C^{\rm op} \to {\rm Cat}.$

Mike Shulman (Mar 08 2024 at 17:53):

I wonder whether we are dealing with an xy problem here. Fabrizio, why do you want to define a presheaf from a lax functor?

Fabrizio Romano Genovese (Mar 08 2024 at 18:10):

This is a good question, and indeed in the end I dropped this avenue of research for something much simpler.
I wanted a presheaf together with some coverage such that the presheaf ended up being a sheaf exactly when the functor was strong/strict. But now I am looking at other stuff to measure the 'how far from being strong you are' problem

Fabrizio Romano Genovese (Mar 08 2024 at 18:11):

As suggested by @fosco , given $F: A \to B$, I tried to play with the functor $B \to Cat$ sending each object to the comma category $F/b$ (or to $b/F$, as one wishes). But as I said, in the end I went for something simpler and dropped (pre)sheaves altogether ^_^