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In order to find a pair of suitable non-isomorphic stacks for which labelings of the coarse space by orders of groups does not distinguish them, we need to find two complex reflection groups satisfying …

SGA1 Exp 13, Vistoli's notes, or the Stacks project). … If you have a morphism $\beta: X \to Z$ in $\mathcal{C}$, and $f: Z \to \mathcal{M}$, then $\beta$ induces a morphism of stacks over $\mathcal{M}$. …

If $G$ is an affine algebraic group, there is a classifying stack $BG$. For any scheme $S$, you have a groupoid $BG(S)$ whose objects are principal $G$-bundles $P \to S$ over $S$ (locally trivial wit …

The three-step process is given as Theorem 3.8 in Ross Street, Two dimensional sheaf theory, J. Pure Appl. Algebra 23 (1982) 251–270 under some smallness assumptions. Without some smallness hypothese …

As Johannes Ebert mentioned in the comments, Noohi has some papers online that describe topological stacks. … For example Grothendieck's Pursuing Stacks is one of the early attempts to apply homotopy theory techniques to work with more abstract objects like $n$-categories and $n$-stacks. …

The category of maps from a test object $T$ to a quotient stack $[X/G]$ has the following general form. Objects are pairs $(P, f)$, where $P$ is a $G$-torsor over $T$, and $f: P \to X$ is a $G$-equiv …

Bhargav said this first in different words, but (by analogy with the homotopy picture) you need your map to be basepoint-preserving. In particular, the point corresponding to the trivial G-torsor sho …

You can find a definition in section 5 of Martin Olsson's paper Logarithmic geometry and algebraic stacks. His web page doesn't have it, but you can find it on Google Scholar. Option 2 works also. …

By standard deformation theory (see e.g., Hartshorne III Ex 4.10, but there are probably better references), the tangent sheaf of $\mathscr{M}_g$ is $R^1\pi_{\ast}(\mathscr{C}, T_{\mathscr{C}/\mathscr …

Here is a class of examples: A sheaf with flat connection on a smooth variety is equivalent to a sheaf on the de Rham stack of that variety, and de Rham stacks of positive dimensional varieties are not … More generally, there are variations on the notion of sheaf with connection, e.g., action of an algebroid, that can be viewed functorially using sheaves on non-algebraic stacks which are stackifications …

Here's a counterexample to your title question, that is not a counterexample to BCnrd's claim: Let $Y =\operatorname{Spec} \mathbb{C}[[t]]$. This scheme has a closed point and an open generic point. …

An object over a scheme $T$ on the left is given by a decomposition of $T$ into a parametrized disjoint union $T_i$ of schemes, and a parametrized family of triples $(x_i, y_i, \phi_i)$, where $x_i$ i …

Stacks are just fibered categories that satisfy good descent properties with respect to some chosen topology. … Notational suggestion: As David Ben-Zvi mentioned in the comments, some people are introducing new terms to distinguish between stacks in categories and stacks in groupoids. …

For the first question, there is an equivalence of stacks between $X/G$ and $*/H$, and Theorem 4.46 of Vistoli's notes gives an equivalence between $H$-equivariant quasicoherent sheaves on a point (i.e …

One class of counterexamples arises from quotients by actions of infinite discrete groups acting freely, but it only works under some definitions of algebraic space. For example, if you specify an ac …