# Questions tagged [grothendieck-topology]

The grothendieck-topology tag has no usage guidance.

120
questions

3
votes

0
answers

173
views

### How to read the definition of Grothendieck Pretopology in SGA4?

In SGA4, the first axiom of a Grothendieck pretopology is given as:
PT0: Pour tout objet $X$ de $C$, les morphismes des familles de morphismes de $Cov(đť‘‹)$
sont quarrables. (Rappelons quâ€™un morphisme ...

6
votes

1
answer

225
views

### Subobject classifier for sheaves on large sites with WISC

Let $\mathsf{C}$ be a possibly large category with a Grothendieck topology satisfying the Weakly Initial Set of Covers condition: there is for each $X$ a set (not a proper class) of covering families ...

8
votes

1
answer

297
views

### The Grothendieck topology of closed immersions on schemes

Let $S$ be a scheme. Let's define a Grothendick topology on $\mathrm{Sch}/S$ where a covering family $\{f_i:Z_i\rightarrow X\}_{i\in I}$ on an $S$-scheme $X$ is a collection of closed immersions of $S$...

1
vote

0
answers

82
views

### 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 Vistoliâ€™s ...

1
vote

1
answer

147
views

### Trivial Grothendieck topology and identity morphisms

So on nLab the definition of a trivial (Grothendieck) topology is the following: "The Grothendieck topology on any category for which only the identity morphisms are covering is the trivial ...

6
votes

1
answer

276
views

### Decategorifying Grothendieck topoi and categorifying topological spaces

(This is in a sense a follow-up to this question.)
I was under the impression these days that Grothendieck topoi were alsoÂą analogous to topological spaces in that the former were left exact ...

6
votes

1
answer

433
views

### Subsheaves of Spec K, K a field

$\DeclareMathOperator\Spec{Spec}\newcommand\Ring{\mathrm{Ring}}\newcommand\op{^\text{op}}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Sh{Sh}$In the category of schemes the objects of the form $\...

4
votes

0
answers

159
views

### 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

164
views

### 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 ...

2
votes

0
answers

139
views

### Is the Grothendieck topology equivalent to its Singleton Grothendieck topology?

I'm using the definition of a Grothendieck topology in Angelo Vistoliâ€™s 2007 Notes on Grothendieck topologies,
fibered categories
and descent theory and I found on nLab about superextensive site, that ...

0
votes

0
answers

118
views

### Why form jointly surjective collections of open embeddings or local homeomorphisms a Grothendieck topology?

$\newcommand{\Top}{\mathit{Top}}\DeclareMathOperator\pr{pr}$In Angelo Vistoliâ€™s 2007 Notes on Grothendieck topologies, fibered categories and descent theory on page 28 we get two examples (2.27 and 2....

3
votes

0
answers

125
views

### Johnstone's Elephant - Lemma C2.1.7 confusion

I don't understand the proof of (ii) in the Johnstone's Elephant:
Lemma 2.1.6 is:
Now consider $\bigcup_{f \in R} f \circ f^*(S)$. This is my notation for the sieve Johnstone references in the proof ...

3
votes

1
answer

366
views

### Proof without sieves: Equivalent grothendieck topologies have the same sheaves

I'm currently learning about sheaf theory with Angelo Vistoliâ€™s 2007 Notes on Grothendieck topologies,
fibered categories and descent theory. And in page 35, there is the following definition of a ...

3
votes

1
answer

198
views

### 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 Vistoliâ€™s 2007 Notes on ...

3
votes

0
answers

186
views

### The etymologies behind certain topologies on the category of schemes

Certain topologies on the category of schemes (or perhaps certain appropriate subcategories thereof) are named rather aptly, e.g. Zariski, Ă©tale, fppf, fpqc, syntomic, smooth, v(aluation), etc., but ...

5
votes

0
answers

157
views

### 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 ...

5
votes

1
answer

230
views

### Group scheme with an isotrivial maximal torus

Let $G$ be a reductive group scheme over a normal ring $A$. Then, we know that Zariski locally it admits a maximal torus.
Let us assume that it admits a maximal torus after a finite surjective (resp. ...

7
votes

1
answer

369
views

### When is a basis of a topological space a Grothendieck pretopology?

Bases of a topological space in point set topology will in general form a coverage on its category of inclusion on open subsets and on its category of inclusion on basic opens, but it takes a bit more ...

4
votes

0
answers

213
views

### Defining the h-topology via v-covers

I have two questions about v-covers and the h-topology (as defined by Voevodsky) which arose when reading Bhatt-Scholze's "Projectivity of the Witt vector affine Grassmannian" available here ...

3
votes

1
answer

200
views

### Sheaves on sites given by a (regular) cd-structure

Let $C$ be a category equipped with a Grothendieck topology generated by a cd-structure (see https://ncatlab.org/nlab/show/cd-structure or Voevodsky's paper Homotopy theory of simplicial presheaves in ...

2
votes

0
answers

483
views

### fppf/ etale Cohomology calculate with Cech cohomology

Let $R$ be a commutative ring with one and $S$ commutative faithfully flat $R$-algebra (that is there is a faithfully flat ring map
let $\phi: R \to S$). Then the so called Amitsur complex
$R \to S^{\...

4
votes

0
answers

279
views

### Building intuition for the étale topology

My Honours supervisors have charged me with building intuition for Ă©tale morphisms and the Ă©tale topology. Their suggestions were to "compute the Ă©tale topology in a few simple cases", such ...

3
votes

0
answers

186
views

### Is the category of covering spaces always a topos?

It is well knows that for a nice (locally path connected, semi-locally simply connected) topological spaces, the category of covering spaces over $X$ is equivalent to the functor category $\left[\Pi_1\...

13
votes

3
answers

1k
views

### 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 ...

6
votes

1
answer

310
views

### Is there a notion of ÄŚech groupoid of a cover of an object in a Grothendieck site?

Given a topological space $X$, and a cover $\mathcal{U} :=\cup_{\alpha \in I}U_{\alpha}$ of $X$, one can define a groupoid called ÄŚech groupoid $C(\mathcal{U})$ of the cover $\mathcal{U}$ by $\...

5
votes

0
answers

169
views

### Étale stack on $\text{Spec}(k)$

Let $\mathcal{F}:\text{Ă‰t}(\text{Spec}(k))^{\mathrm{op}}\rightarrow \text{Set}$ be an Ă©tale presheaf. The Ă©tale sheaf condition for the cover $\text{Spec}(l)\rightarrow \text{Spec}(k)$ is
$$\mathcal{F}...

2
votes

0
answers

189
views

### Localization of Chow groups and flat base change

For any flat morphism $f:X\rightarrow Y$, we have a flat pullback of Chow groups
$$Ch^i(Y)\rightarrow Ch^i(X).$$
A particular example of this is of course an open immersion $U\rightarrow X$. In that ...

2
votes

2
answers

451
views

### What is the geometric description of the set of isomorphism class of $G$-torsors over a site $C$?

Let $X$ be a topological space and $G$ be a topological group. Let $\tilde{G}$ be the sheaf of groups defined by the sheaf of sections of the product $G$ bundle $ \pi_1:X\times G \rightarrow X $.
(1)...

3
votes

2
answers

361
views

### Is any constant Zariski sheaf already a Nisnevich sheaf?

Lat $A$ be a set and $\underline{A}$ the associated constant Zariski sheaf on the category $Sm/S$ of schemes which are smooth over $S$ for a fixed base scheme $S$. Is $\underline{A}$ already a (...

5
votes

0
answers

226
views

### $\ell$-adic Eilenberg-MacLane space and Brown representability

I posted the following question on MathStackexchange, where it was suggested that I should move my question to Mathoverflow, which do here (https://math.stackexchange.com/questions/3550741/algebraic-...

2
votes

0
answers

180
views

### Only discrete topology gives trivial topos?

Given a Grothendieck site $\mathsf{(C,\tau)}$, if the associated Grothendieck topos $\mathsf{Shv(C,\tau)}$ is trivial, i.e. $\mathsf{Shv(C,\tau)}$ consists of the terminal sheaf $*$ only, can we ...

9
votes

0
answers

565
views

### When is fppf better than fpqc (and vice versa)?

Depending on a geometer's needs, they may use the Zariski/etale/syntomic/etc. topology on the spaces they consider. I know some settings where etale topology is better suited for the task than the ...

3
votes

1
answer

267
views

### 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
answer

801
views

### 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{...

2
votes

0
answers

223
views

### Significance of some expected results when defining Grothendieck topology

Let $\mathcal{C}$ be a category. Fixing an object $U$ of $\mathcal{C}$, there are some obvious functors we can associate to it, for example:
$h_U:\mathcal{C}^{op}\rightarrow \text{Set}$ given by $V\...

2
votes

1
answer

300
views

### fppf-extension of algebraic groups is an algebraic group

The problem is the following:
Let $k$ be a field and let $N,G,Q: (\operatorname{Sch}/k)_{\operatorname{fppf}} \longrightarrow \operatorname{Grps}$
be fppf-sheaves from the big fppf-site of $\...

2
votes

1
answer

400
views

### Stack associated to Groupoid object in category $\text{Sch}/S$

Consider the category of manifolds $\text{Man}$.
A groupoid object in the category of manifolds is called a Lie groupoid, denoted by $\mathcal{G}$. There is a way to associate a stack (over the ...

2
votes

1
answer

592
views

### Why care about Grothendieck topology? [closed]

Noah Schweber said here the following:
Why would you want a notion of sheaf theory for objects more general
than topological spaces? Well, the original motivation (to my
understanding) was to ...

4
votes

1
answer

484
views

### 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?
...

4
votes

1
answer

572
views

### Do coherent sheaves on rigid analytic spaces form an abelian category?

It is said that the category of sheaves of abelian groups on a Grothendieck site(topology) is an abelian category. On the other hand, it is known that in usual algebraic geometry, given an variety (...

2
votes

2
answers

566
views

### In the definition of big/small étale/fppf/... site, is their covering set really a set?

My definition of a site is a pair $(\mathcal{C},\DeclareMathOperator{\Cov}{Cov}\Cov (\mathcal{C}))$ where $\mathcal{C}$ is a category and $\Cov (\mathcal{C})$ is a set consisting of elements of the ...

7
votes

1
answer

757
views

### Applications of $h$-topology and $h$-descent

This is a technical problem about applications of Grothendieck topologies. In some recent works, the technique of $h$-topology and $h$-descent is very useful, for an introduction see https://stacks....

0
votes

0
answers

317
views

### 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 Ă©tale site). Is there a simple class of morphisms such that there is a ...

6
votes

0
answers

148
views

### Does an fppf cohomological class annihilated by an etale cover come from etale cohomological group?

Let $X$ be a scheme, $F$ a sheaf on the fppf site of $X$, and $\alpha\in H^i_{\mathrm{fppf}}(X,F)$ such that it is trivialized by an etale cover of $X$. Does $\alpha$ lie in the image of the canonical ...

4
votes

1
answer

177
views

### When can a scheme be recovered from its descent groupoid?

Suppose that $ Y $ is a scheme and $ f\colon X\to Y $ a covering of $ Y $ in some Grothendieck topology on the category of schemes (i.e. if $\{ U_i\to Y\}$ is a covering in the topological sense, then ...

3
votes

1
answer

301
views

### Why are sheaves of a coverage the same as those on its generated Grothendieck coverage?

Preliminaries: There are lots of variations on the settings in which we define sheaves. I am concerned with the details linking the general definition below to the Grothendieck topology it generates; ...

3
votes

0
answers

275
views

### Locality in Grothendieck Topologies

Let $\mathcal{C}$ be a category and $\mathcal{J}$ be a Grothendieck topology on it (i.e., $(\mathcal{C},\mathcal{J})$ is a site). Then what is a good notion of locality in it?
I came up with the ...

2
votes

3
answers

939
views

### Classical point-set topology using Grothendieck topologies

Its well known that the category of opens $O(X) $of a topological space $X$ can be endowed with a Grothendieck topology making it into a site. I am looking for references which take the reader through ...

7
votes

0
answers

205
views

### Pushout of Nisnevich sheaves

Let us consider the projective line $\mathbb{P}^1$ over a field $k$ and take the following open embeddings
$$j_{\epsilon}\colon \mathbb{P}^1\setminus\{0,\infty\} \to \mathbb{P}^1\setminus\{\epsilon\}$$...

10
votes

2
answers

585
views

### 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 ...