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.
I posted this on MSE, but it might have been suitable for MO, not sure, I'd be happy to post it there as well, if it would be well received. Also, not sure what tags to put here, I went with higher-category-theory but noticed there was no condensed-mathematics tag.