All Questions
Tagged with sheaf-theory stacks
26 questions
25
votes
3
answers
5k
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 ...
- 14.7k
16
votes
2
answers
2k
views
Understanding the definition of stacks
First of all I should apologies if this question does not count as a research level one. I asked the same question on MathUnderflow and didn't receive any answer. Let me cross post (copy and paste) it ...
- 1,093
13
votes
1
answer
1k
views
How is a Stack the generalisation of a sheaf from a 2-category point of view?
A stack is usually given in terms of:
-A category $F$ fibered over another $C$ such that the functor $Hom(x,y), x,y \in F(\alpha), \alpha \in C$ is a sheaf
-The descent data are effective.
There ...
- 433
13
votes
1
answer
894
views
What is the difference between internal presheaves and presheaves on a total space?
Suppose that $\mathbb{C}$ is a category with finite limits and that $\mathcal{D}$ is a category internal to $\mathbb{C}$. We can also represent $\mathcal{D}$ as a fibration $\mathbb{D}\to\mathbb{C}$.
...
- 131
11
votes
2
answers
2k
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Finiteness conditions on simplicial sheaves/presheaves
Could someone give an overview, or just some examples, of "finiteness conditions" for simplicial sheaves/presheaves and/or simplicial schemes? Any answer or comment about this would be interesting, ...
- 5,637
11
votes
2
answers
878
views
Does sheafification preserve sheaves for a different topology?
Let $T_1$ and $T_2$ be two Grothendieck topologies on the same small category $C$, and let $T_3 = T_1 \cup T_2$ (by which I mean the smallest Grothendieck topology on $C$ containing $T_1$ and $T_2$). ...
- 66.8k
10
votes
1
answer
777
views
Why is there no stack of $\ell$-adic sheaves on a curve?
One of the main players in the categorical geometric langlands correspondence is the moduli stack of rank n integrable connections on a complex curve. The reason for considering such objects is that ...
- 101
10
votes
1
answer
786
views
Sites which are stacks over themselves
A site C with pullbacks is subcanonical (all representable presheaves are sheaves) if and only if its codomain fibration $Arr(C) \to C$ is a prestack (all hom-presheaves are sheaves). Is there a ...
- 66.8k
10
votes
1
answer
506
views
What is the total space of a stack after all?
From my general experience I think for myself of what follows as some kind of taboo question for some reason: in my imagination, everybody wants an answer to this but somehow thinks it shall not be ...
- 18.4k
10
votes
0
answers
212
views
Does the category of $G$-equivariant sheaves have enough injectives?
The question is related to this one.
Let $k$ be a field and $X$ be a topological space. We consider Sh$(X)$, the category of sheaves of $k$-vector spaces on $X$.
Let $G$ be a topological group which ...
- 9,019
9
votes
1
answer
4k
views
surjective morphism of schemes or epimorphism of sheaves?
I have a technical question coming from reading Toen's master course on stacks.
If we view schemes as locally ringed spaces then there we could define a morphism to be surjective if it the underlying ...
- 1,889
9
votes
0
answers
248
views
How does the machinery of left-exact comonads generalize from sheaves to stacks?
Suppose that we have two Grothendieck sites, their associated sheaves $\mathcal{E}=\rm{Sh}(\bf{C},J)$ and $\mathcal{F}=\rm{Sh}(\bf{D},K)$ and a geometric surjection $f:\mathcal{E}\to\mathcal{F}$. This ...
- 91
8
votes
1
answer
1k
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Fpqc sheafification and localisation
I am slightly confused about sheafification at the moment.
I first learned sheaves defined as a subcategory of presheaves, then I was told that sheaves are also a localisation of presheaves, then I ...
- 1,273
7
votes
1
answer
787
views
Definition of the cotangent complexes of Artin stacks
I am studying the notion of the cotangent complexes of Artin stacks reading LMB's book and Olsson's paper. According to them, the cotangent complexes are defined as projective systems in their derived ...
- 479
6
votes
2
answers
580
views
Associating a principal bundle to a torsor
In Introduction to the language of stacks and gerbes, Moerdijk defines a torsor to be a sheaf $\mathcal{S}$ on $X$ with a freely transitive left-action of a sheaf of groups $\mathcal{G}$, such that $...
- 2,227
6
votes
1
answer
417
views
Etale sheaves on algebraic spaces vs. Etale sheaves on affines
Let's fix a field $k$. First, consider $Aff_k$ to be the category of affine finite type $k$ schemes. On this category, one can define the etale topology and thus consider the site $Aff_k^{et}$, then ...
- 595
6
votes
0
answers
183
views
Dense (∞,1)-subsites
So if $C$ is a 1-site and $D$ is a subsite (with the induced coverage), there are some conditions that ensure that the pre-composition and right Kan extension functors yield an equivalence of ...
- 439
5
votes
1
answer
367
views
Check that a Sheaf is Invertible Etale Locally
A question about following statement from Martin Olsson's book on Stacks. In the proof of Proposition 13.2.9. (p 269) is claimed that certain sheaf $K$ on a nodal curve $C$ is invertible it suffice to ...
- 6,028
5
votes
1
answer
631
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 "B-...
- 1,993
4
votes
2
answers
340
views
Sheafification of presheaf of trivial vector bundles is the stack of vector bundles
This is a deliberately vague question, possibly obvious to experts. Let $F$ be a field. Over the (say, fpqc) site of $F$-schemes, we may define a presheaf $T^{\textrm{pre}}$ that takes a scheme $S$ ...
user333445
3
votes
0
answers
205
views
Category of sheaves of vector spaces on BG
Let $G$ be an affine group scheme over $\mathbb{C}$. I am interested in understanding the differences between different notions of sheaves on the stack $pt/G = BG$. For any algebraic stack $X$ one can ...
- 53
3
votes
0
answers
384
views
Definition of derived category of a stack
In their book, Bernstein an Lunts define the equivariant derived category in several ways. One can be expressed as follows:
Let $X$ be a say complex variety with an action by an algebraic group $G$.
...
- 13.2k
2
votes
1
answer
373
views
Local question and descent category for a quasi-coherent sheaf on $\mathbb{G}_m$-gerbe
Update: I removed what I thought was unecessary and tried to be more straightforward in the hope to get an answer.
Context:
Suppose I have a $\mathbb{G}_m$-gerbe $\mathcal{G}$ over a scheme $X$ with ...
- 65
2
votes
0
answers
158
views
Torsors for nonabelian groups and maps to contracted products
$\newcommand\op{\mathrm{op}}$My question concerns torsors for a sheaf of groups $G$ that is not commutative, and left/right are messing me up. A left $G$-torsor is equivalent to a right $G^{\op}$-...
- 1,094
1
vote
0
answers
137
views
The stack $\operatorname{GL}_2/B$
Let $F$ be the functor from the category of affinoid Tate algebras over $\mathbb{Q}_p$ to the category $\mathrm{Sets}$, which maps an affinoid $\operatorname{Spm} R$ to the set of orbits $\...
- 2,907
1
vote
0
answers
67
views
Is there an inverse image functor for sheaves on stacks?
I'm interested specifically in an inverse image functor between differentiable stacks, ie. stacks coming from Lie groupoids. Specifically, if I have a morphism of Lie groupoids $H\to G$ and I have a ...
- 1,198