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Stream: learning: questions

Topic: geometric occurrences of profunctors

Peva Blanchard (Jan 07 2025 at 21:58):

Let $C,D$ be two categories, $P : D^{op} \times C \to \text{Set}$ a profunctor, and $F$ be a presheaf over $C$. As discussed in the thread on co-yoneda and yoneda lemma, $F$ can be seen as a generalized space.

If I recall correctly, the profunctor $P$ can be used to "migrate" $F$ and obtain a presheaf $G$ over $D$, which is then interpreted as a generalized space, but of a different kind.

Naming $F$ and $G$ generalized spaces has a geometric flavor. Is there a similar thing for profunctors? Or more precisely, is there an application in geometry where some well-known kind of correspondance between spaces turns out to be a profunctor?

Peva Blanchard (Jan 07 2025 at 22:12):

I think I found a similar question and the corresponding answers on math overflow.

Fernando Yamauti (Jan 07 2025 at 22:29):

I don't know exactly how you want to see those things as spaces, but a profunctor induces a functor between presheaf topoi. Sometimes that turns out to be left exact. That means it gives a geometric morphism $\widehat{D} \rightarrow \widehat{C}$. When that happens, $P$ turns into a ($BD$-$BC$)-bibundle that transforms principal $BD$-bundles (you might see it written, instead, as $BD$-local systems sometimes) into principal $BC$-bundles. Maybe we can drop the left exactness, but, then, we would get bundles where the action is not principal. We could also drop groupoid completion and talk about directed homotopy types instead. Is that the sort of thing you are looking for?

fosco (Jan 08 2025 at 08:02):

[[collage]] s of profunctors have a distinct geometric flavour... and a small community of Russian algebraic geometers took this seriously, as they describe the operation of "resolution of singularities" in terms of collages ("gluings") https://arxiv.org/abs/1212.6170

David Corfield (Jan 08 2025 at 08:56):

In the world of $(\infty, 1)$-categories, they're seeing profunctors in this light.

Take a look also at integral transforms on sheaves and a bunch of other related nLab pages.
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Peva Blanchard (Jan 09 2025 at 23:56):

Thank you all for your answers!