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6
votes
1answer
642 views

Basic questions about stacks 2

I have again three basic questions about stacks. 1) When we consider categories fibered in groupoids, do we always mean small or essentially small groupoids? Especially I want to know if algebraic ...
2
votes
1answer
271 views

Irreducibility of quotient stacks.

Let $[X/G]$ be a quotient stack such that $X$ is irreducible and $G$ acts trivially on $X$ (I am just adding automorphisms to every point). Under which hypothesis is $[X/G]$ irreducible as an Artin ...
11
votes
1answer
869 views

Qcoh(-) algebraic stack?

The $2$-functor $\text{Qcoh} : \text{Sch}^{op} \to \text{Cat}$, which sends a scheme to its category of quasi-coherent modules, is a stack by Descent Theory. Is it actually an algebraic stack? If not, ...
9
votes
1answer
583 views

“Approximating” $BGL(1)$ by projective spaces

Given a representation $V$ of a group $G$, we can think of $V$ as a vector bundle over the classifying stack $BG$, and we can define its index $\chi(BG; V)$ to be the dimension of the $G$-invariant ...
5
votes
2answers
453 views

Automorphism groups and etale topological stacks

Recall that an etale topological stack is a stack $\mathscr{X}$ over the category of topological spaces (and open covers) which admits a representable local homeomorphism $X \to \mathscr{X}$ from a ...
1
vote
0answers
488 views

Coarse Moduli space of proper Deligne-Mumford stacks

Let F be a finite type proper Deligne-Mumford Stack over a perfect field. Is it true that the coarse moduli space of F is proper?
5
votes
1answer
576 views

What is the stalk of a stack?

When we study sheaves of sets (on a space X or a site C) we are often interested in the stalks of the sheaf (at either a point $p:1\to X$ or a left exact, cover-preserving functor $a:C\to Sets$). I ...
9
votes
0answers
427 views

Deformations of some simple quotient stacks.

I am interested in stacks of vector bundles on varieties and how deformations of the variety (including non-commutative ones) reflect themselves in deformations of the stack of vector bundles. I will ...
4
votes
1answer
325 views

Ramification formula for orbifolds

It's well known for smooth curves that if $\pi:X\to Y$ is a finite map, $K_X=\pi^*K_Y+Ram(\pi)$, this is just the Riemann-Hurwitz formula at the level of line bundles. I've been told that this ...
10
votes
2answers
769 views

Basic questions about stacks

I'm trying to understand some basics of stacks in algebraic geometry and have three questions: 1) As far as I understand, the moduli stack of vector bundles over a scheme $X$ is a replacement for the ...
13
votes
1answer
487 views

Does a degeneration always have a larger-dimensional automorphism group?

Suppose $\newcommand{\X}{\mathcal{X}}\X$ is an algebraic stack over a field $k$, $\xi$ is a $k$-point which has another $k$-point $x$ in its closure ($x$ is an isotrivial degeneration of $\xi$). ...
6
votes
1answer
623 views

Exactly how is 'the diagonal is representable' used for algebraic stacks…

...apart from stating properties of $(s,t):X_1 \to X_0\times X_0$ for the a presenting algebraic groupoid $X_1 \rightrightarrows X_0$? Once we know that given a stack $\mathcal{X}$ we have a smooth ...
5
votes
1answer
336 views

Categories of descent data

Let us work over the etale site $\mbox{Aff}/S$ (for the sake of definiteness) for some fixed base scheme $S$, where the covers are jointly surjective etale maps $\{ U_i \rightarrow U\}_{i\in I}$ (and ...
5
votes
2answers
733 views

What about stacks of categories in algebraic geometry? II

I've made this a new question, rather than expanding the first one. Torsten gives a good answer, and it partially illustrates in practice the 'second approach' I outlined in my other question. (You ...
10
votes
1answer
434 views

Local structure of Deligne-Mumford stacks

Let $\mathcal{X}$ be a separated Deligne-Mumford stack over an algebraically closed field $k$ and let $X$ be the corresponding coarse moduli space, which we assume to exist. There is a map ...
17
votes
1answer
964 views

What about stacks of categories in algebraic geometry?

Stacks qua moduli spaces were introduced to keep track of nontrivial automorphisms of the objects they parameterize. In essence they are groupoids of objects with some form geometric cohesion. The ...
11
votes
0answers
321 views

Is there a non-quotient stack with affine stabilizers whose good moduli space is a geometric point?

Definitions: One says that a map $\pi\colon\mathcal X\to X$ from an algebraic stack to an algebraic space is a good moduli space if $\pi$ is cohomologically affine and universal for maps to schemes. ...
5
votes
0answers
127 views

Maximal algebraic sub-groupoids

By a theorem of Ehresmann, topological and Lie categories (by which I mean categories internal to $Top$ and $Diff$ respectively, with the condition that the source and target in the latter case are ...
1
vote
1answer
366 views

References for constructible sheaves on complex analytic stacks

I'm looking for references on constructible sheaves and the six operation formalism on analytic stacks (stacks fibered over complex analytic spaces). Does anyone have some suggestions? Basically I ...
7
votes
1answer
724 views

Why is lack of functoriality of the Lisse-Etale topology specific to the Lisse-Etale topology?

I'm trying to follow the explanation given in Olsson's "Sheaves on Artin stacks" for the lack of functoriality for lisse-├ętale topology: Let $f:Y \to X$ be a morphism of algebraic stacks. The functor ...
1
vote
0answers
136 views

$G_m$-cohomology of a motif (that corresponds to a stack?)

As in the question For a G-variety, what could one say about the motif of the corresponding simplicial variety I am in the following situation: $G$ is an algerbraic group, and X is a smooth ...
4
votes
2answers
454 views

Is there a “geometric” language that describes the equivalence groupoid of a foliated manifold?

Sitting on the couch in my office is a certain groupoid. It's waiting for me to say something to it. My problem is that I don't know its language. My question here is for some suggestions. Here, ...
3
votes
2answers
458 views

Sheaves on stacks and interesting functors

Let $G$ be a finite group and $H \subset G$ a normal subgroup. Consider $G$, $H$, and $X=G/H$ as affine algebraic groups over some algebraically closed base field $k$. I hear that there is an ...
2
votes
1answer
263 views

Rational points of an algebraic space over finite field

If $X$ is an algebraic space of finite type over a finite field $k$, then I think it is true that the set of $k$ rational points of $X$ is finite. This is of course true for $X$ is a scheme. I wish ...
5
votes
3answers
680 views

Proving that a map is a morphism

Example : Consider the (open, not compactified) moduli space of stable maps $ \mathcal M_g(1,d)$ of maps of smooth curves of genus $ g$ to $\mathbb P^1$. To each map we associate its branch divisor, ...
9
votes
0answers
458 views

Two questions on algebraic stacks

Question 1: The main reference on algebraic stacks (Laumon and Moret--Bailly) defines a separable algebraic stack as one having universally closed diagonal. For schemes separability is simply defined ...
6
votes
1answer
272 views

The plus construction for stacks of n-types

In Jacob Lurie's Higher Topos Theory, Section 6.5.3, he briefly mentions that to stackify a presheaf of $n$-groupoids, one needs to apply the "+"-construction $\left(n+1\right)$ times, and in general, ...
8
votes
0answers
272 views

What is the signficance of the existence of a moduli stack to a moduli problem?

This is a question in pretty unfamiliar territory for me, so if I have conceptual mistakes please correct me. Let's say we begin with a naive moduli problem: we want a moduli space (whatever space ...
3
votes
1answer
202 views

closed substacks of cartesian powers of a stack

Let $\mathbb{Z}/2\mathbb{Z}$ act on $\mathbb{A}^1$ as $x \mapsto -x$, and let $\mathscr{X}$ be the quotient stack. It has coarse moduli space $\mathbb{A}^1$ and a residual gerbe ...
1
vote
1answer
176 views

When do maps of ineffective orbifolds descend to their effective part?

If $$f:\mathscr{X} \to \mathscr{Y}$$ is a map between (possibly ineffective) orbifolds (in the sense of differentiable stacks, or orbifold groupoids), does it follow that $f$ induces a map between ...
3
votes
0answers
161 views

When is a category of groupoid schemes fibred over schemes?

The category of topological categories $Cat(Top)$ is fibred over $Top$ - the functor sending a groupoid $X_1 \rightrightarrows X_0$ to its object space $X_0$ is a Grothendieck fibration. Now one can ...
5
votes
2answers
219 views

What condition on a “bibundle between categories” generalizes “right-principal bibundle between groupoids”?

My question is long on background and motivation, and almost but not quite answered over at the nLab. I'll write up a bunch before asking my question (feel free to skip to the end or look at the ...
14
votes
2answers
828 views

Applications of Stacks

I've been aware of stacks since grad school, and I can usually follow in rough lines a discussion about stacks, but I've often wondered what particular (purely!) scheme-theoretic argument or theorem ...
2
votes
1answer
301 views

Weak colimits of weak and strict presheaves in groupoids

Let $C$ be a small category, and for this question, let groupoid mean an (essentially small) groupoid. There are two 2-categories in question: the 2-category of strict presheaves in groupoids and ...
5
votes
1answer
741 views

Is every (Artin/DM) algebraic stack fibered in sets an algebraic space?

If $X$ is an algebraic stack fibered in sets (and therefore essentially a sheaf), is it an algebraic space? It seems conceivable that at least when $X$ is Deligne-Mumford, it is actually an algebraic ...
5
votes
1answer
334 views

Can Deligne-Mumford stacks be characterized by their restriction to a small subcategory?

If I have a Deligne-Mumford stack $\Pi : X \to (\mathrm{Sch}/k)$ for some field $k$, can it be reconstructed from $\Pi^{-1}(C) \subset X$ for some "small" subcategory $C \subset (\mathrm{Sch}/k)$? For ...
6
votes
1answer
481 views

Classifying stacks and homotopy type of a point

Suppose we are working in a category of schemes over a scheme $S$. The scheme $S$ itself is geometrically a ``point''. Let $G$ be a group scheme that acts on a scheme $X$. The quotient stack $[X/G]$ ...
5
votes
1answer
349 views

Who first came up with the idea of essential/Morita equivalence of internal groupoids/categories?

The idea that stacks can be identified with groupoids internal to the base site $S$ up to what is variously called essential/Morita equivalence is well known. The basic idea is that one takes the ...
6
votes
3answers
537 views

Fibered category with an adjoint inclusion

Suppose $X:D \to C$ is a fibered category (I do not assume the fibers to be groupoids). Suppose that $X$ is actually left adjoint to a fully faithful embedding $C \hookrightarrow D$. Is there a ...
4
votes
1answer
389 views

Intrinsic Characterization of when an orbifold (or more general stack) is effective?

Recall that an orbifold is an etale and proper differentiable stack $X$. Etale means that it admits an etale atlas $M \to X$ from a manifold $M$ (which is to say it is represented by an etale Lie ...
6
votes
1answer
659 views

When is the K-theory presheaf a sheaf?

Let $F$ be a Deligne-Mumford stack that is of finite type, smooth and proper over $\mathrm{Spec~}k$ for a perfect field $k$. Consider $K_m$, the presheaf of $m$-th $K$-groups on $F_{et}$, the etale ...
15
votes
3answers
1k views

stacks as Morita equivalence classes

I have often encountered definitions of the kind "stacks are equivalence classes of groupoids under Morita equivalence" in topological or differentiable context, with the notion of Morita equivalence ...
7
votes
1answer
588 views

Are root stacks characterized by their divisor multiplicities?

Definitions/Background Suppose $S$ is a scheme and $D\subseteq S$ is an irreducible effective Cartier divisor. Then $D$ induces a morphism from $S$ to the stack $[\mathbb A^1/\mathbb G_m]$ (a ...
3
votes
0answers
849 views

When is the base change morphism an isomorphism?

This is a rewrite of a previous question, which was in turn a follow up question to Leray-Hirsch principle for étale cohomology The motivation is to clarify some points in Torsten Ekedahl's ...
3
votes
1answer
481 views

Question about global quotient stacks

In "Brauer groups and quotient stacks", Edidin et. al prove the following theorem: Theorem 2.7. Let $\mathcal{X}$ be an algebraic stack over a Noetherian base (of finite type). Then the diagonal ...
4
votes
1answer
271 views

Noether-style isomorphism theorem for stacks?

Let $G$ be a group, actiong on a set $X$ and $H$ a normal subgroup. Then we have a canonical isomorphism $$(X/H)/(G/H)\rightarrow X/G$$ I would like to have a statement like this for stacks, more ...
10
votes
0answers
666 views

Seeing stacks in the Calculus of Functors

Recently I was told (by an algebraic geometer) that when algebraic geometers look at the Calculus of Functors, they think of stacks. When I look at the Calculus of Functors, I see a categorification ...
7
votes
3answers
523 views

Twisted curves, admissible covers, and an algebraic analogue of a specific monodromy computation

This problem arose when trying to understand the stack of twisted stable maps into a stack (specifically BG), as introduced by Dan Abramovich, Angelo Vistoli and several co-authors (Olsson, Graber, ...
3
votes
1answer
282 views

Notion of stack fibered in monoidal categories?

This can be seen as a follow up my question here: Is there a notion of "fibered category with boxproducts"? Given a monoidal fibration $f:E\rightarrow B$ (i.e. a strict monoidal functor ...
15
votes
2answers
805 views

Applications of topological and diferentiable stacks

What are some examples of theorems about topology or differential geometry that have been proven using topological/differentiable stacks, or, some examples of proofs made easier by them? I'm well ...