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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 …

1.) It's possible to define stacks on ANY category equipped with a Grothendieck topology (such a category with a topology is called a site). In particular, this holds true for the Zariski site. Moreov …

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 …

I will try to answer this question in a way relevant to more than one field, however, to be honest, I'm rather unconventional in the sense that my experience in this area stems from topological and di …

Suppose I have a functor $f:(C,J)\to(D,K)$ between Grothendieck sites. Is there a condition on $f$ such that $f_!$ (the left adjoint to $f^*$) sends "$J$-epimorphisms", to $K$-epimorphisms, where by $ …

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 …

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 …

I think you get the same sheaves if and only if your topos of sheaves can be expressed as as sheaves on some singleton pretopology: If $V \stackrel{f}{\rightarrow} X$ has the property that there exis …

Let $S$ be your Grothendieck site. What you want is a subcategory $j:B \hookrightarrow S$ such that the Grothendieck topology of $S$ restricts to $B$ in the sense that every covering sieve of $b \in B …

If $X_1 \rightrightarrows X_0$ is a groupoid and $\mathcal{X}$ is the associated stack, consider the atlas $p:X_0 \to \mathcal{X}$. If you form the weak $2$-pullback of $p \times p:X_0 \times X_0 \to …

It is known that the category of schemes is not cocomplete (e.g. see this question: Colimits of schemes). However, do diagrams of schemes for which every morphism is etale have colimits? More generall …

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 …

$\newcommand\Top{\mathit{Top}}\DeclareMathOperator\Sh{Sh}$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 …

I believe I have the answer in the setting of sheaves of sets. Let us first do this for presheaves, $Set^{C^{op}}$. This category is Cartesian-closed. This can be seen by setting $Y^X(U):=Hom(X \time …

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 …