Does [condensed / pyknotic mathematics](https://ncatlab.org/nlab/show/condensed+set) afford an (yet!) another approach to orbifold theory?


Let me admit up-front that I don't know much about either condensed / pyknotic mathematics or about orbifold theory. But I think the following is a precise question which gets at the the above vague question.


I believe there are several approaches to defining what an orbifold is, so for safety let me restrict attention to the $(2,1)$-category $\mathcal C$ of *topological* orbifolds $X$ such that $X$ is the orbifold quotient of a topological manifold $X\_0$ by a properly discontinuous group action. I hope that even if there are differences between definitions of topological orbifolds, they should all at least agree on the category $\mathcal C$ up to equivalence of $(2,1)$-categories.


On the other hand, every compactly generated weak Hausdorff space, and in particular every topological manifold, may be regarded as a condensed set in a canonical way. Moreover, there inclusion functor $Set \to Gpd$ induces an inclusion functor $Cond(Set) \to Cond(Gpd)$ from condensed sets to condensed groupoids. Let $\mathcal D$ be the full, $(2,1)$-subcategory of $Cond(Gpd)$ whose objects are $(2,1)$-categorical quotients of topological manifolds by properly discontinuous group actions.


There are canonical inclusion functors $Man \to \mathcal C$ and $Man \to \mathcal D$ where $Man$ is the category of topological manifolds (where every 2-morphism is the identity).


**Question:** Is there an equivalence of $(2,1)$-categories $\mathcal C \simeq \mathcal D$ respecting the two inclusions $Man \rightrightarrows \mathcal C, \mathcal D$?



Condensed / pyknotic approach to orbifolds?
Great question! I think the answer ought to be yes. But I must also make a disclaimer that I don't really know what orbifolds are, and the comments by David Roberts make be believe that what I thought they should be is not what they actually are.


To me, an orbifold should be to a manifold as a Deligne--Mumford stack is to a scheme. In particular, they should naturally form a $2$-category (just like groupoids do). Recall that a Deligne--Mumford stack is a stack (=sheaf of groupoids) $X$ on the category of schemes equipped with the étale topology such that there is a surjective étale map $Y\to X$ from a scheme $Y$. Now the most general definition of an étale map of stacks is actually a bit subtle. Let me only consider the case of $0$-truncated maps (which is the only case relevant here, as $Y$ and thus $Y\to X$ is $0$-truncated). The case of separated étale maps is OK: A map $f: Y\to X$ is separated étale if and only if for all maps $X'\to X$ from a scheme $X'$, the fibre product $Y'=Y\times\_X X'$ is representable by a scheme, and $Y'\to X'$ is separated étale. To verify that this is a good definition, one uses that separated étale maps of schemes satisfy fpqc descent. For general étale maps, one has to iterate this procedure a bit: One should only ask that $Y'$ is an algebraic space, and $Y'\to X'$ an étale map of algebraic spaces. To define the latter, one takes an étale cover of $Y'$ and asks that the composite map to $X$ is étale.


Now the translation to (topological, say) manifolds is somewhat complicated by the fact that separatedness assumptions are much more commonly assumed by default. For example, I think $\mathbb R$ with a doubled origin (i.e. two copies of $\mathbb R$ glued along $\mathbb R\setminus\{0\}$) does not count as a topological manifold, even while its algebraic analogue has long become a honorable member of the category of schemes. I will thus assume that the "correct" notion of orbifold is the one one obtains by pretending that all schemes are separated, and all étale maps are separated étale. One then gets the following notion.


Consider the category of topological manifolds $\mathrm{Man}$, and make it into a site by using open covers as covers. Then an orbifold is a stack (=sheaf of groupoids) $X$ on $\mathrm{Man}$ such that there is a surjective separated étale map $Y\to X$ for some manifold $Y$. Here, a map $Y\to X$ of stacks on $\mathrm{Man}$ is separated étale if after pullback $X'\to X$ from any manifold $X'$, the fibre product $Y'=Y\times\_X X'\to X'$, $Y'$ is a manifold, and $Y'\to X'$ is a local isomorphism.


Here is a different way to proceed, starting from condensed groupoids. Recall that these are sheaves of groupoids on the site $\mathrm{ProFin}$ of profinite sets, with covers given by finite families of jointly surjective maps. Again, we first define separated étale maps $f: Y\to X$ of condensed groupoids: This means that after any pullback $S\to X$ to a profinite set $S$, the fibre product $Y\times\_X S$ is representable by a locally profinite set, and the map to $S$ is a local isomorphism. Any manifold $M$ defines a condensed set $\underline{M}$ via sending $S$ to the set of continuous maps from $S$ to $M$, and this embeds manifolds fully faithfully into condensed sets. Now one can define an orbifold to be a condensed groupoid $X$ that admits a surjective separated étale map $\underline{M}\to X$ from the condensed set corresponding to a manifold $M$.


We claim that these definitions are equivalent. First, there is a functor: Any orbifold $X$ in the condensed sense defines a sheaf of groupoids on $\mathrm{Man}$ by sending $M$ to the groupoid of maps $\underline{M}\to X$.



> 
> Proposition. This functor is an equivalence of categories.
> 
> 
> 


Let me just explain the key step. Assume that $M$ is a topological manifold, and $f: Y\to \underline{M}$ is a separated étale map of condensed sets. Then $Y=\underline{N}$ for some manifold $N$, where $N\to M$ is a local isomorphism. To prove this, the idea is the following. Take any $y\in Y$ mapping to $m\in M$. We want to find a neighborhood of $y$ mapping isomorphically to a neighborhood of $m$. But for separated étale maps, sections always extend from closed subsets (like $\{m\}\subset M$) to small strict neighborhoods (and uniquely so, up to further shrinking the neighborhood). By descent, this can be checked after taking surjections from profinite sets to closed neighborhoods, and after pullback to profinite sets, the claim is easy to see.


Let me also advertise that in [Etale cohomology of diamonds](https://www.math.uni-bonn.de/people/scholze/EtCohDiamonds.pdf), I'm very much proceeding along such lines to define various "geometric objects" in $p$-adic geometry akin to (pretty wild forms of) orbifolds, by essentially adopting the condensed point of view (but this paper predates the introduction of the "condensed" term). In that paper, the role of profinite sets is played by "strictly totally disconnected perfectoid spaces", which are essentially profinite sets of geometric points. The difference here is that there is not just one kind of geometric point, but rather one for any complete algebraically closed nonarchimedean field $C$ (of characteristic $p$). One then studies all geometric objects by regarding them as quotients of (disjoint unions of) such strictly totally disconnected perfectoid spaces. The analogue of the key step is Lemma 15.6 (or really Lemma 11.31) there, comparing étale maps defined via descent to strictly totally disconnected spaces to étale maps defined more geometrically. That it was possible to develop the whole machinery from this point of view of descent to strictly totally disconnected spaces is what convinced me personally that it's not a completely crazy idea to regard manifolds as quotients of locally profinite sets.


