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$\newcommand\Top{\mathit{Top}}\DeclareMathOperator\Sh{Sh}$One advantage is that it gives you a geometric representation for slice topoi of sheaves over a space: Given a topos $E$, $E$ is equivalent to …

Yes, this is just a basic fact in category theory, if interpreted correctly. For $C$ any category, and $F$ any preheaf on $C,$ $F$ is the colimit in presheaves of the diagram $C/F \to C \stackrel{y}{\ …

The way I have always used the word étale in reference to a possibly-not-representable morphism of Deligne-Mumford stacks $f:\mathscr{X} \to \mathscr{Y}$ is that for any étale morphism $X \to \mathscr …

Let $X$ be a scheme over $\mathbb{C}$. When does the topological space $X\left(\mathbb{C}\right)$ of $\mathbb{C}$-points have the homotopy type of a finite CW-complex? When does the topological spa …

Let $X$ be a scheme of finite type over $\mathbb{C}$ and let $Z \hookrightarrow X$ be a closed subscheme. Consider the associated closed inclusion $Z_{an} \hookrightarrow X_{an}$ between their analyti …

Let $X$ be a stack of $n$-groupoids on the site of affine schemes over a fixed base, with the etale topology. If $n=1$ then for $X$ to be Deligne-Mumford, aside from having an etale atlas from an alge …

There is a full and faithful embedding of the category of schemes into the $2$-category of (edit: stricly Henselian ringed) topoi, which sends each scheme $X$ to the topos $Sh\left(X_{et}\right)$ (in …

There is a canonical equivalence of $2$-categories $$St\left(Man/M\right) \simeq St\left(Man\right)/M$$ between stacks on the large site of $M$ and stacks on the site of manifolds equipped with a ma …

I have stumbled upon some literature on Deligne-Mumford stacks, and it seems to me, at least superficially, that there is a strong link between DM-stacks which have "trivial generic stabilizers" and " …

First a couple corrections. If $M$ is represented by quotient stack, it better be a contravariant functor, and moreover, it should probably take values in groupoids, not set. Anyway, here's what going …

I could be wrong, but I am going to be brave and assume that coarse moduli spaces are defined in the analogous way as for topological stacks. If this turns out being incorrect, I will remove this answ …

Local questions: 1) Given a commutative ring $A,$ is $Sh_\infty\left(Spec(A)\right)$ hypercomplete? 2) Given a commutative ring $A,$ is $Sh_\infty\left(Et\left(A\right)\right)$ hypercomplete, where …

I am not an expert in the algebraic category, however, I know that 2) is an open problem in the differentiable category; it is not known if every smooth orbifold is a global quotient stack. It is true …

It depends on what you mean by an étalé "space". As long as $C$ has a small set of topological generators (i.e. as long as $Sh(C,J)$ isn't too large to be a topos), there always exists a certain versi …

There is a classical result which says that the assignment $$M \mapsto C^{\infty}\left(M\right)$$ is an embedding of the category of (paracompact Hausdorff) smooth manifolds into the opposite category …