Questions tagged [grothendieck-topology]

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Are the injections of a coproduct a cover in the canonical pretopology?

Assume we're in a category $C$ with all pullbacks and finite coproducts. Recall that the canonical coverage of $C$ is the finest Grothendieck (pre) topology for which all representables are sheaves. A ...
Joey Eremondi's user avatar
6 votes
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216 views

Generalizing uniform structures as Grothendieck topologies

Recently, I was reading a classical book "Sheaves in Geometry and Logic" by S. MacLane and I. Moerdijk, and then it stroke me that, that the definition of Grothendieck Topology bears some ...
Nik Bren's user avatar
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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....
Joey Eremondi's user avatar
2 votes
1 answer
128 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)/...
Joey Eremondi's user avatar
3 votes
0 answers
188 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 $Cov(𝑋)$ sont quarrables. (Rappelons qu’un morphisme ...
Joey Eremondi's user avatar
6 votes
1 answer
240 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 ...
Rylee Lyman's user avatar
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8 votes
1 answer
317 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 $S$...
FNH's user avatar
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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 ...
Muster Maxfrau's user avatar
2 votes
1 answer
178 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 ...
Saegusa's user avatar
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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 also¹ analogous to topological spaces in that the former were left exact ...
Emily's user avatar
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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 $\...
Nico's user avatar
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4 votes
0 answers
179 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 ...
Sergey Guminov's user avatar
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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 ...
Muster Maxfrau's user avatar
3 votes
0 answers
153 views

Is the Grothendieck topology equivalent to its Singleton Grothendieck topology?

I'm using the definition of a Grothendieck topology in Angelo Vistoli’s 2007 Notes on Grothendieck topologies, fibered categories and descent theory and I found on nLab about superextensive site, that ...
Muster Maxfrau's user avatar
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136 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 ...
Emilio Minichiello's user avatar
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1 answer
423 views

Proof without sieves: Equivalent grothendieck topologies have the same sheaves

I'm currently learning about sheaf theory with Angelo Vistoli’s 2007 Notes on Grothendieck topologies, fibered categories and descent theory. And in page 35, there is the following definition of a ...
Muster Maxfrau's user avatar
3 votes
1 answer
215 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 Vistoli’s 2007 Notes on ...
Muster Maxfrau's user avatar
3 votes
0 answers
188 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, étale, fppf, fpqc, syntomic, smooth, v(aluation), etc., but ...
Dat Minh Ha's user avatar
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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 ...
saolof's user avatar
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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. ...
prochet's user avatar
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7 votes
1 answer
422 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 ...
saolof's user avatar
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4 votes
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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 ...
Hinter's user avatar
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1 answer
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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 ...
Josefien K's user avatar
3 votes
0 answers
551 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^{\...
user267839's user avatar
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4 votes
0 answers
305 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 ...
Martin Skilleter's user avatar
3 votes
0 answers
206 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\...
Chetan Vuppulury's user avatar
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 ...
Sofía Marlasca Aparicio's user avatar
6 votes
1 answer
324 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 $\...
Adittya Chaudhuri's user avatar
5 votes
0 answers
169 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}...
curious math guy's user avatar
2 votes
0 answers
212 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 ...
curious math guy's user avatar
2 votes
2 answers
467 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)...
Adittya Chaudhuri's user avatar
3 votes
2 answers
375 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 (...
Lao-tzu's user avatar
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5 votes
0 answers
229 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-...
curious math guy's user avatar
2 votes
0 answers
192 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 ...
Lao-tzu's user avatar
  • 1,856
10 votes
0 answers
651 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 ...
user avatar
3 votes
1 answer
283 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 ...
John Pardon's user avatar
  • 18.3k
19 votes
1 answer
837 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{...
Oscar Cunningham's user avatar
2 votes
0 answers
225 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\...
Praphulla Koushik's user avatar
2 votes
1 answer
340 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 $\...
Jrodri26's user avatar
  • 123
2 votes
1 answer
470 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 ...
Praphulla Koushik's user avatar
2 votes
1 answer
640 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 ...
Praphulla Koushik's user avatar
4 votes
1 answer
556 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? ...
Heer's user avatar
  • 1,007
4 votes
1 answer
610 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 (...
Hang's user avatar
  • 2,719
2 votes
2 answers
643 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 ...
Z Wu's user avatar
  • 340
7 votes
1 answer
831 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....
sawdada's user avatar
  • 6,148
0 votes
0 answers
319 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 ...
user avatar
6 votes
0 answers
151 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 ...
Heer's user avatar
  • 1,007
4 votes
1 answer
178 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 ...
Gaussler's user avatar
  • 295
3 votes
1 answer
326 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; ...
Tyler Bryson's user avatar
3 votes
0 answers
284 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 ...
Chetan Vuppulury's user avatar