Questions tagged [grothendieck-topology]
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125 questions
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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 ...
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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 $\mathscr{U} = \{...
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What is a topos?
According to Higher Topos Theory math/0608040 a topos is
a category C which behaves like the
category of sets, or (more generally)
the category of sheaves of sets on a
topological space.
Could one ...
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2
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Flat versus étale cohomology
Although the definition of étale ($\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” to ...
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Coverages that are not pretopologies
A coverage on a category $C$ is a collection of covering families $\{u_i \to a\}$ for each object $a$ of $C$ such that for each arrow $b\to a$ there is a covering family for $b$ which fits into a ...
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Topos associated to a category
For each topos $\mathbb E$ let $\mathcal O(\mathbb E)$ be the locally presentable category of objects in $\mathbb E$. We can make $\mathcal O$ into a contravariant functor to the category of locally ...
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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 ...
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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 ...
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6
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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....
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1
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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)/...
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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 ...
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A bestiary of topologies on Sch
The category of schemes has a large (and to me, slightly bewildering) number of what seem like different Grothendieck (pre)topologies. Zariski, ok, I get. Etale, that's alright, I think. Nisnevich? ...
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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 ...
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1
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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 ...
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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 $S$...
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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 ...
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1
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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 ...
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1
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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 ...
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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 $\...
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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 ...
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Cohomologie Etale
Is there an English translation available for Deligne's Cohomologie Etale (Arcata) that is now part of the SGA 4 1/2 ?? Atleast for the first two sections - Grothendieck Topologies and Etale Topology.
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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 ...
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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 ...
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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 ...
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1
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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 ...
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1
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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 ...
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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.
...
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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 ...
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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 ...
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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 "...
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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. ...
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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 ...
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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 ...
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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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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^{\...
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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 ...
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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\...
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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 ...
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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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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
$$\mathcal{F}...
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Localization of Chow groups and flat base change
For any flat morphism $f:X\rightarrow Y$, we have a flat pullback of Chow groups
$$Ch^i(Y)\rightarrow Ch^i(X).$$
A particular example of this is of course an open immersion $U\rightarrow X$. In that ...
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Grothendieck topology for a non-small category
To define a Grothendieck topology of a category, we usually require that the category is small.
Question 1: Why do we need to require the category to be small?
I thought that the problem was that ...
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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\...
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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)...
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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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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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$\ell$-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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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 ...
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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{...
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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 ...
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