# Questions tagged [grothendieck-topology]

The grothendieck-topology tag has no usage guidance.

101
questions

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votes

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### 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 ...

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vote

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146 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

179 views

### Building intuition for the étale topology

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

**3**

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92 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\...

**10**

votes

**3**answers

700 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 ...

**4**

votes

**1**answer

252 views

### Is there a notion of Čech 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 Čech groupoid $C(\mathcal{U})$ of the cover $\mathcal{U}$ by $\...

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156 views

### Étale stack on $\text{Spec}(k)$

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

**2**

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133 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

295 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

290 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 (...

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212 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-...

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121 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

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286 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

206 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 ...

**14**

votes

**1**answer

641 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

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211 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

196 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

228 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 ...

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vote

**1**answer

418 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

340 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

453 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

323 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 ...

**6**

votes

**1**answer

502 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....

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287 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 étale site). Is there a simple class of morphisms such that there is a ...

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votes

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140 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

160 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 ...

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votes

**1**answer

209 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; ...

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233 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

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708 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

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169 views

### Pushout of Nisnevich sheaves

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

**6**

votes

**1**answer

334 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 ...

**10**

votes

**1**answer

446 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 ...

**0**

votes

**1**answer

152 views

### If $J$-coverings can be glued $I$-locally is $J$-locality an $I$-local property? (Reducing descent problems to simpler ones)

Let $(C,J)$ be a category with a grothendieck topology. For every object $X \in C$ there's (I hope) a little site which is the full subcategory of the slice category $C_{/X}$ whose objects are the ...

**9**

votes

**1**answer

700 views

### Are all Grothendieck topologies on Set equivalent?

The category $\textbf{Set}$ can be given a Grothendieck topology where the covering families are jointly surjective families of set inclusions $\{X_i\stackrel{\phi_i}{\hookrightarrow} X\}\in\mathrm{...

**7**

votes

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250 views

### Reference for the Brown-Gersten property for smooth manifolds

A classical result by Brown and Gersten says that to verify the homotopy descent property for the Zariski topology
it suffices to verify it for Zariski squares and the empty cover of the empty scheme.
...

**7**

votes

**1**answer

363 views

### Is there a way to “puncture” a topos?

Let $E$ be a (Grothendieck) topos, e.g. $E = \text{Sh}(X)$ for a topological space $X$. And let $p = (p^*, p_*):\text{Set}\to E$ be a point of $E$, is there a way to "puncture" $E$ in some sense? By "...

**2**

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**1**answer

110 views

### Exercise on “locality” in topos theory

Let $\mathcal E=\mathsf{Sh}(\mathsf C,J)$. Let $A\rightarrowtail \Omega$ be a fixed subobject. For each $X$ in $\mathcal E$, define $T_A(X)$ to be a set of subobjects of $X$ as follows. $U\...

**16**

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**3**answers

1k views

### Voevodsky's Triangulated Categories of Motives and their Relationships

As we know, Voevodsky constructed several candidates for the triangulated category of motives using different constructions and topologies (h, qfh, etale, and Nisnevich).
I would like to know what ...

**20**

votes

**1**answer

671 views

### Can we just use effective descent morphisms (pure morphisms) as covers?

There are a number of notions of "cover" for a scheme: etale, faithfully flat, fpqc, fppf, Zariski, Nisnevich, etc. Most of these have a nice property, which is that a cover of that type satisfies ...

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votes

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518 views

### Grothendieck topologies on $\mathbb{C}$

I would like to consider three sheaves $\mathcal{C}^0$, $\mathcal{H}$ and $\mathcal{S}$ on $\mathbb{C}$ (endowed with the euclidean topology): the first is the sheaf of continuous $\mathbb{C}$-valued ...

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1k views

### Flat versus etale cohomology

Although the definition of etale ($\ell$-adic) cohomology is scary, I have at least some intuition for how it should behave: for instance, when it makes sense, I expect that it should be ``similar'' ...

**10**

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362 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 ...

**3**

votes

**1**answer

226 views

### Sections of morphisms up to fppf covering

Let $f:X\to S$ be a finite type affine morphism of schemes where $S$ is an integral noetherian affine regular scheme whose function field is of characteristic zero.
Assume that all geometric fibers ...

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vote

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247 views

### Isomorphism in a derived category of chain complexes with rational coefficients

Let $C$ be the category of quasi-projective schemes over a base field $k$ (maybe I will need some assumptions on $k$). Let $Ab(C_{\tau})$ be the category of Abelian sheaves on a site $C_{\tau}$, where ...

**46**

votes

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5k views

### What is the purpose of the flat/fppf/fpqc topologies?

There have been other similar questions before (e.g. What is your picture of the flat topology?), but none of them seem to have been answered fully.
As someone who originally started in topology/...

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386 views

### Reference for cdh topology

Let $f:X\rightarrow Y$ be a proper surjective morphism over some base scheme $S$ of finite type, suppose $f$ restricts to an isomorphism over some open $U$ of $X$, we also suppose both $X$ and $Y$ are ...

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votes

**2**answers

420 views

### 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 ...

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votes

**3**answers

1k views

### Gerbes and Stacks

The definition of a gerbe on a smooth manifold that I know is that - after fixing an open cover $U_i$, a gerbe consists of the data of line bundles $L_{ij}$ on two-fold-intersections $U_{ij}$, ...

**12**

votes

**1**answer

1k views

### Reference request: Book of topology from “Topos” point of view

Question: Is there any book of topology in the modern language of topos theory?
Motivation:
In "Sheaves in Geometry and Logic" Mac Lane and Moerdijk say: "For Grothendieck, topology became the ...

**3**

votes

**0**answers

435 views

### Two functorial definitions of schemes

I have been reading a bit about the "functor of points" theory for schemes. There seem to be two ways of going about defining schemes this way:
Equip the category $\textbf {Psh}=\operatorname{Fun}(\...