I have a few questions, but they're not properly formulated just yet, but they stem from a few simple facts :
- In homotopy theory, the homotopy hypothesis postulates that topological spaces (up to homotopy) and ($\infty$) groupoids should really be the same thing. There are many ways to make this statement precise, let me state the one I have in mind here : There is a Quillen equivalence $(\left\vert- \right \vert \dashv \mathrm{Sing}):sSet_{Quillen} \rightleftarrows Top$, where $sSet_{Quillen}$ denotes the category of simplicial set with its usual model structure where the Kan Complexes / groupoids are precisely the cofibrant-fibrant objects. The adjunction is given by geometric realisation of a simplicial set and the singular complex of a topological space. This Quillen equivalence induces an equivalence of $(\infty,1)$-categories $Top \simeq \infty$–$Grpd$. In that sense, topological spaces and $\infty$-groupoids really are the same.
- Dustin Clausen & Peter Scholze's recent work on condensed mathematics aiming to reunite algebra and topology led to the following idea : condensed sets are a good notion to replace topological spaces. Condensed set can be defined in a few equivalent ways, as sheaves on the site of compact Hausdorff spaces, or compact Hasdorff totally disconnected spaces or compact Hausdorff extremally disconnected spaces where coverings in each case are given by finite families of jointly surjective maps. These are a nice replacement for topological spaces for various reasons, but the word replacement is justified by the following remark : most "nice" topological spaces can be considered as a condensed set, indeed for a space $X$ (for instance a $CW$-complex), one can define $\underline{X}:ExtrDisc^{op}\to Set$ via $\underline{X}:=Hom_{Top}(-,X)$, and this actually gives a fully-faithful embedding of $CW$ complexes into $Cond(Set)$ the category of condensed sets.
With all this in mind, it seems we have to very different notions that aims to be stand ins for topological spaces, one up to homotopy, and the other, up to homeomorphism I think. My first question is the following : surely we have a comparison of these two ? I certainly don't expect these two approach to be equivalent, for one, in groupoids, topological spaces are really considered up to homotopy, but I'm pretty sure that homotopy equivalent spaces wouldn't give the isomorphic condensed set, and I don't know if condensed sets have a weaker notion equivalence than iso, and this lead to my next questions. Have there been approach to homotopy theory internally to condensed sets ? I don't know if this question really makes sense, but I suppose one way to be more precise would be the following : have people investigated potential model category structure on condensed sets ? (I suppose the same questions works for condensed …, where you can replace … with whatever you prefer, abelian groups, rings, etc). I suppose the general theme of my questions is the following idea: on the hand we have a quite established theory of higher categories and infinity groupoids, higher topos theory, with for instance all of Lurie's work. On the other hand, we have a new theory of condensed mathematics, with Clausen & Scholze's work. I'm sure people have already started mixing the two, and I'm very curious as to how that was done, but I'm not sure where to look for. Maybe there's a way to make sense of "condensed infinity groupoid" and maybe the homotopy hypothesis would be that these classify/are equivalent to condensed sets up to homotopy in a way (seeing as condensed sets are a "replacement" for topological spaces).
[This post was originally posted on MSE, but after some thought it probably should have been posted here in the first place, and a comment suggested to post it here as well.]
Best Answer
The way in which "condensed sets are similar to topological spaces" is very different from the way in which "$\infty$-groupoids are similar to topological spaces". In fact, condensed mathematics is, in a sense, a refutation of the homotopy hypothesis. Let me try to explain that.
Classically, topological spaces play at least two very distinct roles : there is the "topology = study of shapes, donuts, etc." role where the local topology is rather boring (it's all based on $\mathbb R$ more or less) and it's the globality of the space that is relevant (the circle is two copies of $\mathbb R$, put together in an interesting way); and there is the "topology = way to put more structure/restrictions on some kind of infinitary structure to control it" role, where now the local topology and the global topology can both be very nontrivial. An example of this is the $p$-adic topology, which is not very relevant from the "donut" perspective on topology, but is extremely important for many arguments in algebra.
Homotopy theory deals with the former, where the local triviality of the topology, and the relaxed notion of equivalence allows you to turn most things into essentially combinatorics, which is, in a sense, why this can be faithfully encoded by $\infty$-groupoids, or, model-dependently, simplicial sets.
Condensed math aims to deal with the latter : its central purpose is to be able to equip algebraic gadgets with some kind of "topological structure" that allows you to control even very infinitary situations, just in a better way than topological spaces can (they do not mush so well with algebraic structures).
This is why condensed people can also freely merge the two approaches and consider "condenses anima" : these two "directions" of topology are essentially unrelated, and so you can face a situation where you have both directions, sort of orthogonally. An example of that would be if you want to consider derived $p$-adic things, where you have a homotopy theory direction ("derived", or maybe chain complexes and so on) and a "topology", or better, "condensed" direction (the $p$-adic topology); or maybe derived (real or complex) functional analysis with now the homotopy direction still, and the condensed direction is less algebraic but still relevant.
Overall, the point is to try to distinguish the two. In particular, one can try to do homotopy theory of condensed sets and so on and put model structures and compare that to anima, but from this perspective it might be misguided : the condenswd direction is trying to capture something fundamentally different from the homotopy direction - it's in that sense that condensed math is some kind of refutation of the homotopy hypothesis : "topological spaces" (condensed sets) are not the same as homotopy types (anima, $\infty$-groupoids), and should not be mistaken for one another. And this perspective now allows one to appreciate condensed anima as a third thing.