# Questions tagged [sites]

For questions related to the mathematical notion of site, which among others generalizes the notion of topological space. For questions on internet-sites, use online-resources, but note that these questions need to be very specific to research-level mathematics to be on-topic.

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### 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....

2
votes

1
answer

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### 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)/...

6
votes

0
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### Non-invertible map between principal bundles only locally trivial in a supercanonical Grothendieck topology?

It is a truth universally acknowledged that a map between principal bundles must be in want of an inverse.
However, the construction of said inverse in the context of a more general site $(S,J)$ ...

8
votes

1
answer

441
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### Tensor product of sites

Let $C, D$ two Grothendieck sites. Since the corresponding toposes $E, F$ are locally presentable categories, then (by Gabriel-Ulmer duality) they correspond to limit theories $X, Y$ (that is, small ...

1
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0
answers

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### Why is the induced singleton pretopology closed under pullbacks?

Let $(C,\mathcal{T})$ be an arbitrary site with a pretopology $\mathcal{T}$. The category $C$ has coproducts.
As a pretopology I mean the definition 2.24 of Grothendieck topology
in Angelo Vistoli’s ...

2
votes

1
answer

184
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### Necessary and sufficient conditions for all sheaves on a site to be continuous functors?

All representable functors are continuous. This makes it possible to associate additional natural operations with them, which are absent for arbitrary presheaves.
What are the sufficient and what are ...

4
votes

0
answers

199
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### Geometric intuition for pf in fppf-topology

I am trying to learn a bit about algebraic spaces and the various topologies on categories of schemes in general. And, as it seems to always be the case, I am struggling with understanding how exactly ...

0
votes

1
answer

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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 ...

1
vote

0
answers

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### Reference Request: Cech cohomology of complexes on an arbitrary site

I am looking for a reference which is equivalent to this stacks project page [1], except formulated in the generality of an arbitrary site. I checked the "Cohomology on Sites" section of the ...

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2
answers

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### 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 ...

1
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0
answers

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### Extending Grothendieck topology from analytic manifolds to spaces?

Let $k\text{-Man}$ denote the Euclidean site of $k$-analytic manifolds where $k=\mathbb{R}, \mathbb{C}$. In words, $k\text{-Man}$ is the usual category of real/complex analytic manifolds considered ...

3
votes

1
answer

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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 ...

5
votes

0
answers

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### Is there something similar to Lawvere-Tierney topologies for Abelian categories?

Lawvere-Tierney topologies generalize the notion of local operators on a Topos from Sheaf toposes over a Grothendieck site to arbitrary Toposes.
However, while the special case of Sheaves of sets or ...

3
votes

0
answers

487
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### 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 ...

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0
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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
votes

0
answers

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### Does Fpqc sheaf over category of rings imply representability

I am trying to read the article "Algebrization and Tannaka duality" by Bharghav Bhatt. In Corollary 1.2, he says that given a qcqs algebraic space $X$ (I am interested in the case when $X=\...

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3
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### Resources for topos theory

I am trying to learn topos theory and I am finding a strong scarcity of resources. Is there any canonical textbook to refer someone to when learning this topic?
So far, I have only been able to find ...

2
votes

0
answers

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### 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 ...

1
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0
answers

178
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### Confusion in understanding the notion of $G$ Principal bundle where $G$ is a geometric group over a site

The first paragraph of the section Overview in the paper Principal infinity-bundles - General theory by Nikolaus, Schreiber and Stevenson https://arxiv.org/abs/1207.0248 precisely reads the following:
...

2
votes

0
answers

643
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### Čech-Alexander complex in computing (crystalline/prismatic) cohomology

I have a naive question about Čech-Alexander complexes in prismatic cohomology (although I suspect that the situation is similar for crystalline cohomology).
They seemed to be introduced as a method ...

7
votes

0
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353
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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 ...

25
votes

2
answers

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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, ...

3
votes

1
answer

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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 ...

19
votes

1
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### Has this "backwards" perspective on toposes been studied?

Topos theory can be seen as a categorification of topology via the following analogies.
\begin{array}{|c|c|}
\hline
\text{locales}&\text{Grothendieck toposes}\\\hline
\text{open sets}&\text{...

3
votes

1
answer

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### "Covering-flat" part in definition of morphism of sites

Let $(\mathcal{C},\mathcal{I})$ and $(\mathcal{D},\mathcal{J})$ be sites where $\mathcal{C}, \mathcal{D}$ are categories and $\mathcal{I}$ and $\mathcal{J}$ are Grothendieck topologies on $\mathcal{C}...

4
votes

1
answer

609
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### How to construct cup-product in a general site?

Let $\mathcal{C}$ be a site on the category of schemes for some Grothendieck topology, $X$ a scheme, and let $F,G$ be two sheaves of abelian groups on $\mathcal{C}$. Do we have cup-product as follows?
...

0
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0
answers

319
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### Grothendieck topology on a scheme equivalent to the circle

Suppose a class of morphisms of schemes is reasonable enough so that we can associate a small site to any scheme (just like small étale site). Is there a simple class of morphisms such that there is a ...

12
votes

1
answer

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### Points of the big Zariski site

It's relatively simple to show that the geometric morphisms $ \mathbf{Set} \to \mathrm{Sh}(\mathbf{CRing}^\mathrm{op}_{\mathrm{fp}}, \mathrm{Zar})$ correspond to local rings.
More precisely, since ...

2
votes

0
answers

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### 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?

10
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2
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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 ...

10
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0
answers

750
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### 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 ...

4
votes

0
answers

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### Is the restriction of sheaves from the big to small etale sites an equivalence?

Let $X$ be a scheme. Let $(\mathrm{Sch}/X)_{ét}$ and $X_{ét}$ be the big and small étale sites, resp. Is the restriction functor from $\operatorname{Sh}((\mathrm{Sch}/X)_{ét})$ to $\operatorname{Sh}(...

6
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0
answers

176
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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 ...

4
votes

1
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### Stacks for the extensive topology?

Recall that any extensive category can be canonically endowed the structure of a site via the extensive topology, which is the Grothendieck topology whose covering morphisms are the coproduct ...

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0
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### Morphism of 2-sites

Let $X$ and $Y$ be Grothendieck sites. A $\textit{morphism}$ between $X$ and $Y$ is a functor on the underlying categories that is covering-flat and preserves covering families. See https://ncatlab....

5
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0
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### 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,...

5
votes

1
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### Categorification of covering morphisms

Given a category $\mathsf{A}$, let $\mathsf{Fam}(\mathsf{A})$ be its free coproduct cocompletion (which is always extensive). This means every object has a unique up to iso presentation as a coproduct ...

1
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0
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### $\mathbb E$-descent maps in topological spaces in terms of different sites?

The paper Facets of Descent I by Janelidze and Tholen defines $\mathbb E$-descent maps as those for which $\Phi^p:\mathbb EB\longrightarrow \mathsf{Des}_\mathbb{E}(p)$ is an equivalence of categories.
...

0
votes

1
answer

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### A confusion about covering flatness

I'm reading this entry on nLab. But I'm confusing with the notion of covering-flatness. More precisely, I meet some trouble when I try to show that the $Sets$-valued flatness is a special case of ...

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0
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458
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### 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 ...

3
votes

2
answers

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### Morphism on schemes induced by continuous morphism of sites

I am beginner in the theory of Grothendieck topologies and I have the following question.
Let $X, Y$ be schemes over an algebraically closed field $k$. Denote by $X_{et}$ and $Y_{et}$ the Etale sites ...

7
votes

1
answer

340
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### Which dense inclusions of sites are ∞-dense?

An inclusion of sites f: D→C is dense if it induces an equivalence between the categories of sheaves on C and D.
Likewise, f is ∞-dense is it induces an equivalence between the ∞-categories of ∞-...

3
votes

2
answers

520
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### 1st cech cohomology groups on ringed sites

Let $(C, O)$ be a ringed site -- i.e., $C$ is a small category with a grothendieck topology $\tau$ and $O$ a sheaf of rings on the site $(C,\tau)$. In this context, for any object $U$ of $C$ one can ...

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1
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### Points in sites (etale, fppf, ... )

I asked a part of this in an earlier question, but that part of my question didn't receive precedence.
Etale site is useful - examples of using the small fppf site?
Let $X$ be a scheme (assume it ...

27
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2
answers

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### Etale site is useful - examples of using the small fppf site?

Edit: After the answers and comments, I'm hoping for a little bit of elaboration (in the comment to the answer below.) Also, question 2 was discussed here:
Points in sites (etale, fppf, ... )
There, ...