Anyone into Rigid cohomology? Gotta get into it, looks tough...

>>8230480
I hardly know any crystalline cohomology yet. It looks pretty hardcore, or maybe just new. The nLab page is lacking, so I will pirate a book and dig around a bit. I just want to figure out what the ambient ∞-topos is...

>>8230480
Just starting to learn algebraic topology. Can't help you. Sorry.

>>8230499
OP here, what can you tell about approaching geometry/concepts via \infty-topoi ?Are you alg. topologist at first?

>>8230499
libgen.io >> git bukz

Anyone has a word to give about topoi?

>>8230536
Topos is just a category with some special properties, which makes it Set-like in many respects.

>>8230506
Basically, cohomotopy abstractly just looks at contravariant hom functors in an (∞,1)-category. You get cohomology after applying the connected component functor. When the category is a topos and the representing object is nice, you get the usual properties of cohomology.

Conversely, homotopy looks at covariant homs before applying the connected component functor.

The category itself determines a cohomology theory. For integral cohomology one looks at the prototypical (∞,1)-topos Top, and the representing object is the Eilenberg-Maclane space for the integers.

Sheaf cohomology looks specifically at homs of sheaves. This generalizes vertically to ∞-stacks. This is of course at the core of Grothendieck's revolution of algebraic geometry.

>>8230555
Oh, also, if the representing object has certain structure, that structure may lift to the cohomology or homotopy sets. So, in the case of using pointed spheres for homotopy groups, the cogroup structure on spheres lifts to a group structure on homotopy sets. This is why the choice of representing object matters: the more free structure we get, the more tools we have to study objects. If we used the point • for cohomology, the cohomology sets lacks so much structure that we can't seperate any two objects using just that. The universal coefficient theorem ends up just saying that the Eilenberg-MacLane space for the integers is as complex as a coefficient object needs to be; everything more complex produces redundant information about the space.

In this setting, the ∞-Yoneda lemma is very useful: we may determine a space entirely by the homotogy it represents, or by the cohomotopy it represents. The failure of cohomology being sufficient to seperate everything is due to the connected component functor I mentioned. We just look at connected components in the hom spaces rather than the higher homotopies, so we are only able to ask about homotopy types. Theoretically, with a sufficiently highly-structured object to give us the tools for efficiently using all of the data of hom spaces, we could determine things up to homeomorphism using homotopy and cohomotopy. This is not really necessary, but it is there.

>>8230557
Any readable ref for "folklore" about this? like a review with main theiorems, ideas, and without so much technical details.

>>8230596
The page on the nLab titled "motivation for sheaves and cohomology" or something to that tune is a nice start, and then the page "cohomology" is rather extensive. As you said, it is mostly folklore, so there isn't a whole lot of focused research on it. Baez and Shulman have some lecture notes called "Lectures on n-Categories and Cohomology," which is a delightful read and also what first got me hooked on this mode of thinking. Schreiber has some posts on the nCafé, too.

>>8230639
Reading your nLab link.
https://ncatlab.org/nlab/show/motivation+for+sheaves,+cohomology+and+higher+stacks

For me, with étalé space of a sheaf, any section is a function to a space over your base, so sheaves were a relevant notion of "mesurable things over your space". My personal pov was though the comparaison where sections of a sheaf are like a measurement machine (somewhere) on your space. Sheaf of probe is close and nice as well!

>>8230678
Yep, all good stuff! My philosophy has been that maps /into/ a thing in some sense measure local data, and then maps /out/ of a thing measure global data in some way. We embed something into something bigger to get what it looks like overall, and we probe it to get a local view.

In fact, I am working on formalizing all of this. I am looking in particular at localizations and globalizations, how they interact, and how they can be constructed from simple input data in a reasonable way.

>>8230693
Really sounds like what I'm working on hahaha, working on this cohomology now. And your philosophy is interesting, glad to read it.

>>8230707
Thanks, I'm glad to have someone show interest! Happy studies, anon.

>>8230718
You made my day ;) See you in some confs, anon