# dg.differential geometry – Condensed / pyknotic approach to orbifolds?

Does condensed / pyknotic mathematics 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 $$Man rightrightarrows mathcal C, mathcal D$$?