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1
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1answer
178 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
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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
223 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 ...
15
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2answers
861 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
303 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
756 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
342 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
532 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
365 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
541 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
405 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
678 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 ...
16
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 ...
8
votes
1answer
627 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
890 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
497 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
281 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 ...
12
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0answers
688 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
563 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
288 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
812 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 ...
11
votes
0answers
532 views

Stacks in the fpqc topology

This is related to Matt Satriano's earlier question about an analog of Artin's theorem for stacks with an fpqc cover by a scheme. Suppose one developed the theory of stacks in the fpqc topology and ...
4
votes
1answer
360 views

Does the concept of a basis for a topology on a category exist?

If we want to define a sheaf F on a topological space X and we have a basis B for the topology of X, what we can do is to define objects and restrictions for guys in B, check that they satisfy the ...
6
votes
2answers
505 views

Universal property of X//G?

Given an operation of say a topological group on a topological space, one can form the quotient stack X//G: the stack associated to the action groupoid. Does this stack satisfy some kind of universal ...
6
votes
1answer
570 views

When is a stack (NOT) geometric?

Following the terminology of $n$-Lab, a geometric stack $\mathcal{X}$ on a site $\mathcal{(C,J)}$ is a stack for which there exists a representable epimorphism $X \to \mathcal{X}$ from an object $X$ ...
9
votes
1answer
386 views

Double Category of Topological Stacks

There are two equivalent ways of describing topological stacks. One is the "stacky" definition, that is, a topological stack is a stack $\mathbb{X}$ on $Top$ (a Grothendieck universe thereof, if ...
17
votes
6answers
1k views

Categorical construction of the category of schemes?

The answer to the following question is probably well known or the question itself is well known not to have a reasonable answer. In the latter case could you please let me know what the "right" ...
2
votes
4answers
4k views

Cotangent bundle of a differentiable stack

If you ever wanted to construct the tangent bundle of a differentiable stack, it's relatively simple: First, if $\mathbf{X}$ is a stack coming from a Lie groupoid $\mathcal{G}$, you could just say ...
12
votes
2answers
836 views

Ordinary cohomology of stacks

Let $\mathbf{X}$ be a stack over $Top$ (a lax sheaf of groupoids, or some such thing). If it admits a surjective representable map $F \to \mathbf{X}$ then one can form the iterated fibre product to ...
5
votes
1answer
354 views

Connections on principal bundles via stacks?

Let G be a Lie group and M a smooth manifold. Suppose that P is a principal G-bundle over M. Then by Yoneda, this corresponds to a smooth map $p:M \to \left\[G\right]$, where $\left\[G\right]$ is the ...
4
votes
0answers
440 views

finite etale covering of stacks

If $Y \to X$ is a finite etale map of schemes, then there exists a finite Galois morphism $Z \to X$ (i.e. it's a $Aut(Z/X)$-torsor) that factors as $Z \to Y \to X.$ The case when $X$ is normal is easy ...
7
votes
2answers
1k views

Stacks in the Zariski topology?

I have two naive questions about stacks. 1) Is it possible to define stacks in the Zariski topology? Presuming you can: 2) If a stack has a coarse moduli, and the coarse moduli space is a ...
5
votes
1answer
254 views

What is the local structure of a Lie groupoid?

A manifold is locally $\mathbb R^n$. An orbifold is locally $\mathbb R^n/\{\text{finite group}\}$. Is there a similar way to think about the local structure of a Lie groupoid $G_1 \rightrightarrows ...
8
votes
2answers
1k views

Why do gerbes live in H^2 ?

Line bundles on a scheme $X$ live in $H^1(X,O_X^*)$, where $O_{X}^{*}$ is the sheaf of invertible functions. If $X$ is noetherian separated, then we can think of this $H^1$ to be Cech cohmology ...
7
votes
0answers
395 views

Conditions for “bootstrapping” a smooth DM stack?

In the preprint "Smooth toric DM stacks", Fantechi, Mann and Nironi define the stacks of their title, and show that each of these can be obtained through the following sequence of steps: 1) start ...
7
votes
3answers
635 views

Positivity in stack geometry

I was wondering how much of the theory say of Lazarsfeld books can be carried to algebraic stacks (if this has been done). Do we have a sensible notion of an ample (big, nef) line bundle? Of an ample ...
21
votes
3answers
1k views

What can we do with a coarse moduli space that we can't do with a DM moduli stack?

A couple weeks ago I attended a talk about the Keel-Mori theorem regarding existence of coarse moduli spaces for Deligne-Mumford stacks with finite inertia. Here are some questions that I have been ...
7
votes
5answers
1k views

Representation of Groupoids

The title is vague, my actuall question is the following: Has the representations of groupoids been systematically studied? Is there any new phenomenon, compare with the representation of groups? ...
12
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1answer
1k views

In what topology DM stacks are stacks

Background/motivation One of the main reason to introduce (algebraic) stacks is build "fine moduli spaces" for functors which, strictly speaking, are not representable. The yoga is more or less as ...
7
votes
3answers
422 views

Degrees of etale covers of stacks

This is probably pretty basic, but as I said before I'm just beginning my way in the language of stacks.Say you have an etale cover X->Y of stacks (in the etale site). Is there a standard way to ...
19
votes
2answers
722 views

Different interpretations of moduli stacks

I'm taking my first steps in the language of stacks, and would like something cleared up. The intuitive idea of moduli spaces is that each point corresponds to an object of what we're trying to ...
20
votes
1answer
2k views

fpqc covers of stacks

Artin has a theorem (10.1 in Laumon, Moret-Bailly) that if $X$ is a stack which has separated, quasi-compact, representable diagonal and an fppf cover by a scheme, then $X$ is algebraic. Is there a ...
5
votes
1answer
211 views

how good an approximation to the equivariant derived category is given by the Grassmannian filtration of the classifying space?

So, let's say one has an action of $GL_n$ on an algebraic variety $X$ over a field $k$, and two objects $F,G$ in the equivariant derived category (i.e., the derived category of constructible sheaves ...
3
votes
1answer
427 views

Approximation of stacks / algebraic spaces

Let $B$ be a ring which is the colimit of rings $B_\lambda$. Let $X_\lambda$ be a stack (not necessarily algebraic) over $B_\lambda$ such that $X_\lambda \times_{B_\lambda} B_\mu = X_\mu$ and let $X ...
16
votes
3answers
2k views

Stacks and sheaves

I'm a bit confused by the double role which sheaves play in the theory of stacks. On the one hand, sheaves on a site are the obvious generalization of a sheaf on a topological space. On the other ...
3
votes
2answers
650 views

Is there a good notion of `Separated Stack'?

A scheme is separated if the diagonal inclusion $X \to X \times X$ is a closed immersion. I what to know if there is a good generalization of `separated' for algebraic stacks? My usual stack ...
4
votes
1answer
880 views

What is a proper stack?

I have seen the use of the word "proper Deligne-Mumford stack". Now, it is clear to me what it means for a morphism f of stacks to be proper: as usual it should be representable, and every morphism ...
8
votes
1answer
660 views

coarse moduli space of DM stacks

This is related to another one of my questions on DM stacks. In Brian Conrad's article 'The Keel-Mori Theorem via Stacks', a sufficient condition on for an Artin stack to have coarse moduli space is ...
5
votes
3answers
889 views

Is the inertia stack of a Deligne-Mumford stack always finite?

Let X be a DM stack over a field k. We follow the definition in Laumon and Moret-Bailly's book, so that its diagonal is quasi-compact (and hence diagonal is of finite type). Then is the diagonal ...
17
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2answers
826 views

Algebraic versus Analytic Brauer Group

Let $X$ be a smooth projective algebraic variety over $\mathbb{C}$. Then I think that someone (Serre?) showed that the Cohomological Etale Brauer Group agrees with the torsion part of the Analytic ...