Questions tagged [grothendieck-topology]
The grothendieck-topology tag has no usage guidance.
125 questions
4
votes
1
answer
283
views
Are the two definitions of fppf topology on the category of schemes the same?
Consider the definition of fppf (pre)topology on the category of schemes $\mathrm{Sch}$.
Maybe, most textbooks define an fppf covering of $U\in\mathrm{Sch}$ as a family of morphisms $\mathscr{U} = \{...
- 125
7
votes
1
answer
101
views
Bisimplicial spaces as a coequalizer of maps between "simpler" bisimplicial spaces
From a bisimplicial space $T$, one can consider the simplicial spaces $p \mapsto T_{pp}$, $p \mapsto | q \mapsto T_{pq}|$, and $q \mapsto |p \mapsto T_{pq}|$, where $| \cdot|$ denotes geometric ...
- 71
2
votes
1
answer
94
views
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 ...
- 237
7
votes
0
answers
271
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 ...
- 519
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
3
votes
0
answers
215
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 ...
- 237
7
votes
1
answer
255
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 ...
- 1,996
8
votes
1
answer
342
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 $S$...
- 329
1
vote
0
answers
95
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 Vistoli’s ...
- 119
2
votes
1
answer
213
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 ...
- 173
6
votes
1
answer
343
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 also¹ analogous to topological spaces in that the former were left exact ...
- 11.8k
6
votes
1
answer
455
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 $\...
- 775
4
votes
0
answers
222
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 ...
- 1,071
0
votes
1
answer
177
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 ...
- 119
3
votes
0
answers
172
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 ...
- 119
3
votes
0
answers
142
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
513
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 ...
- 119
3
votes
1
answer
244
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 ...
- 119
3
votes
0
answers
191
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 ...
- 1,516
5
votes
0
answers
213
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 ...
- 1,947
5
votes
1
answer
260
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. ...
- 3,472
7
votes
1
answer
465
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 ...
- 1,947
4
votes
0
answers
266
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 ...
- 313
3
votes
1
answer
225
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 ...
- 33
3
votes
0
answers
641
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^{\...
- 6,028
4
votes
0
answers
342
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 ...
- 721
3
votes
0
answers
226
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\...
- 438
13
votes
3
answers
2k
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
343
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 $\...
- 1,215
5
votes
0
answers
170
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
$$\mathcal{F}...
- 4,418
2
votes
0
answers
237
views
Localization of Chow groups and flat base change
For any flat morphism $f:X\rightarrow Y$, we have a flat pullback of Chow groups
$$Ch^i(Y)\rightarrow Ch^i(X).$$
A particular example of this is of course an open immersion $U\rightarrow X$. In that ...
- 4,418
2
votes
2
answers
490
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)...
- 1,215
4
votes
2
answers
416
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 (...
- 1,906
5
votes
0
answers
232
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-...
- 4,418
2
votes
0
answers
212
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 ...
- 1,906
11
votes
0
answers
743
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 ...
user145520
3
votes
1
answer
315
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 ...
- 18.7k
19
votes
1
answer
884
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{...
- 3,137
2
votes
0
answers
228
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\...
- 6,023
3
votes
1
answer
384
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 $\...
- 133
2
votes
1
answer
559
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 ...
- 6,023
2
votes
1
answer
678
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 ...
- 6,023
4
votes
1
answer
659
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?
...
- 997
6
votes
1
answer
670
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 (...
- 2,789
2
votes
2
answers
731
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 ...
- 452
7
votes
1
answer
977
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....
- 6,622
0
votes
0
answers
325
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 ...
user138661
6
votes
0
answers
152
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 ...
- 997
4
votes
1
answer
180
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 ...
- 295