All Questions
Tagged with sheaf-theory sites
18 questions
1
vote
0
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73
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Why can we not convert GATs / EATs / limit sketches to sites?
I think I'm in the process of understanding something very subtle here, and I could use an expert's double check. So basically, my question is whether what I write is correct.
(Non-finitary) GATs, ...
- 511
6
votes
1
answer
395
views
Relationship between canonical topology on a topos and its site of definition
The canonical (Grothendieck) topology for a category $C$ is the largest (finest) topology such that every representable presheaf over $C$ is a sheaf.
According to First Order Categorical Logic Lemma 1....
- 237
2
votes
1
answer
151
views
Is the slice of a subcanonical site also subcanonical?
A subcanonical site is one for which every representable functor is a sheaf.
For a subcanonical site $C$, the fundamental theorem of topos theory says that there is an equivalence $Sh(C/c)\cong Sh(C)/...
- 237
0
votes
1
answer
177
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Does the (Vistoli-)sheafification induce isomorphism?
Given a presheaf, in Angelo Vistoli's 2007 Notes on Grothendieck topologies,
fibered categories
and descent theory
there is a construction of the sheafification (Proof for theorem 2.64).
Note: In ...
- 119
19
votes
2
answers
393
views
Why is $1$ not a dense sub-site in a group with the trivial Grothendieck topology?
A friend of mine had the following question while reading the section "C2.2 The topos of sheaves" in "Sketches of an Elephant".
Let $G$ be a group (considered as a category with ...
- 6,962
3
votes
1
answer
244
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Compatibility of pullbacks with an equivalence relation
This question was originally posted last week in Math Stack Exchange (see here).
I'm currently working on the proof of the existence of the sheafification in Angelo Vistoli’s 2007 Notes on ...
- 119
3
votes
0
answers
530
views
Flasque sheaves on a site
This is a cross-post from MathStackexchange.
We define a flasque sheaf on a site as one whose first Čech cohomology vanishes for every covering of every object of the site. I know this definition is ...
- 153
9
votes
0
answers
308
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Refinement of hypercovers by ordinary covers
I am asking for references and discussions of statements of the form
Every bounded hypercover can be refined by an ordinary cover
By "bounded" I mean "finite height". E.g., are ...
- 4,519
2
votes
0
answers
296
views
Small sheaves on big sites
Background: If one works with sheaves on small etale site over a fixed scheme (which is really an essentially large category), one can instead work with sheaves on the affine etale site (which turns ...
- 319
7
votes
0
answers
362
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What is a morphism of ∞-sites?
Recall that a morphism of sites
is a covering-flat functor
that preserves covering families.
Morphisms of sites can be identified with those
geometric morphisms of induced toposes
for which the ...
- 37.8k
25
votes
2
answers
3k
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Are there (enough) injectives in condensed abelian groups?
The question is very simple : does $Cond(\mathbf{Ab})$, the category of condensed abelian groups (as defined in Scholze's Lectures in Condensed Mathematics), have enough injectives ?
Does it, in fact, ...
- 15.9k
3
votes
1
answer
315
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What to call a morphism of sites inducing an equivalence on categories of sheaves?
Is there a standard term for a morphism of sites $(C,J)\to(C',J')$ which induces an equivalence on sheaf categories $\operatorname{Sh}(C',J')\xrightarrow\sim\operatorname{Sh}(C,J)$? The nLab entry ...
- 18.7k
2
votes
0
answers
264
views
Pullback of a constant sheaf over arbitrary sites
Given a geometric morphism between arbitrary Grothendieck topoi, $f:\mathcal{Sh(D)}\to\mathcal{Sh(C)}$, does the pullback $f^{-1}$ (i.e, the left adjoint) take constant sheafs to constant sheafs?
- 187
11
votes
2
answers
664
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Equivalence of the definitions of a sheaf in SGA4 and in "Categories and Sheaves"
I asked this question on Mathematics Stack Exchange, but got no answer.
I don't understand why the definition of a sheaf (Definition 17.3.1 (ii)) given in the book
[KS] Categories and Sheaves by ...
- 4,416
10
votes
0
answers
762
views
Differential Forms in Infinite Dimensions
In Kriegl/Michor's book "The convenient setting of global analysis", they define the space of differential $k$-forms on a possibly infinite-dimensional manifold $M$ as the space of smooth sections of ...
- 9,853
6
votes
0
answers
183
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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
0
answers
448
views
Examples of nonstable ∞-categories in which sifted colimits commute with finite limits
What are some natural examples (if any) of nonstable ∞-categories in which finite limits commute with sifted colimits (or rather just colimits over Δ^op)?
Stable ∞-categories do satisfy this property,...
- 37.8k
13
votes
0
answers
481
views
Making the conceptual leap from locales to Grothendieck topologies?
I find the definition for locales and sheaves on locales to be straightforward, but I'm stumbling over the idea of a Grothendieck topology. Is there a nice way to see roughly how the latter ...
- 373