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Questions tagged [grothendieck-topologies]

For questions on the Zariski-open immersion, étale, flat, or other topologies defined via classes of morphisms in a category of schemes. Not to be confused with the usual Zariski topology in commutative algebra and algebraic geometry.

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Drinfeld upper half plane admissible open

I have a possibly silly question about the Drinfeld upper half plane. It is "well-know" that if $K$ is a complete local field then $\Omega_K = \mathbb{P}^1(\mathbb{C}_k) \backslash \mathbb{P}^1(K)$ ...
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Pullback of sheaves topology and sites

Let $T = Top$ be the site of topological spaces with the usual open covering and let $\mathcal{F}$ be a sheaf on the site $T$. Naturally, for a space $X \in Ob(T)$, the sheaf $\mathcal{F}$ induces a ...
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Alternative formulation of Grothendieck topology

In Mac Lane and Moerdijk's Sheaves in Geometry and Logic there is a reformulation of the Grothendieck topology conditions in terms of arrows, namely a Grothendieck topology on a small category $\...
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1answer
59 views

Constant sheaves on the étale site of a scheme

I am learning étale cohomology with Tamme's book. When talking about example of abelian sheaves on the étale site, he mentions the following equality for an abelian group $A$ : let $A_X$ be the ...
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1answer
32 views

Grothendieck topology on stacks/fibred category

Let $p : \mathcal{S} \rightarrow \mathcal{C}$ be a stack (or fibred category, I do not know if just being a fibred category is enough) over the site $\mathcal{C}$. I have read somewhere (I forgot ...
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138 views

Kummer sequence etale topology

Consider the category $C=Sch/S$ of schemes over $S$ and let $n \in \Gamma(S,\mathcal{O}_S)^{*}$. It is possible to show that $$0 \rightarrow \mu_{n,S} \rightarrow \mathbb{G}_m \rightarrow \mathbb{G}_m ...
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Is there a fiber bundle/espace-etale interpretation of sheaves on a Grothendieck site

Whenever I'm doing sheaf things and I have a construction that involves sheafifying, I find it convenient to think "the thing that has the same stalks but sections must be locally trivial sections to ...
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1answer
38 views

(Pre)sheaf epimorphism admits a section?

Let $f: F\to G$ be an epimorphism of (pre)sheaves of sets on a Grothendieck site. Does it admit a section? By this I mean a morphism $g: G\to F$ with $f\circ g={\rm id}$. I'm particularly interested ...
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50 views

Sheaf Axiom for Presheaves on Sites

My question concerns a statement about two equivalent (why ?) characterisations of sheaves on sites introduced in https://en.wikipedia.org/wiki/Grothendieck_topology#Sites_and_sheaves We start with a ...
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106 views

Set theoretic issues in the definition of a site in Stacks Project

I've been learning about sites from the Stacks Project, which is generally very precise in its terminology, but I've found some of their conventions very confusing in this part. Their definition of a ...
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1answer
86 views

Grothendieck Universe

I have a question concerning the Grothendieck's universe: Let fix a GUniverse $U$. Is a $U$-set the same as a $U$-category or is there a subtle difference?
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What changes in the sheaf theory of topological spaces with the “étale topology”?

The customary site structure on the category of topological spaces has covering families given by open covers. What "happens" if we refine this topology and let any jointly surjective family of local ...
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Map from schemes to stacks

I have just started studying stacks. Trying to understand the theory I was thinking about a (very interesting) toy example: $ BG $ the classifying stack of a smooth (over a base scheme $ S $) group G. ...
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Is it known that the etale topology on schemes does not have cd structure?

Is it known that the etale topology on schemes does not have cd structure? If yes, can you please give me some reference.
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1answer
68 views

The étale topos is coherent: does the scheme need to be quasicompact?

It is known that the little étale topos of a scheme $X$ is coherent, i.e. the Grothendieck topology of the étale site is generated by a basis of finite coverings. For example, Butz and Moerdijk say ...
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Grothendieck sites based on large categories?

A Grothendieck site is defined as a small category with certain structure. My question is: does it really have to be set-based? Are there notions of Grothendieck sites which are class-based? Would ...
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“Coverings” and “covers” in Grothendieck topologies

Consider a topological space $X$, and an open cover of $X$. Now when choosing a Grothendieck topology we select "coverings", which in the case of $\mathcal O(X)$ should be open covers. But not every "...
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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 ...
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Coincidence of classical notion of covering in classical topology and covering sieve in grothendieck topology

Let $X$ a topological space, $\mathcal{O}(X)$ the usual category of the open sets of $X$ and $(\mathcal{O}(X),J)$ any site. Do we necessarly have for each open set U and for each sieve in $J(U)$, $(...
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1answer
29 views

Minimal site that induces a stack from a psuedofunctor

I'm working with Vistoli's definition of stacks [1] Let $\Phi:C^{op}\to {Cat}$ be a psudeofunctor. Is there always a minimal Grothendieck topology on $C$ such that it induces a stack? If not, are ...
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88 views

When sheafification functor is open

Let $\mathcal{G}$ be a Grothendiek topos and let $(\mathcal{C},J)$ be a site for $\mathcal{G}$. Is it true that if the sheafification functor $$ \mathcal{G} \leftrightarrows \text{Set}^{\mathcal{C}^...
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Borel structure on Grothendieck topology, sort of …

Given a usual topological space $X$ we define in a natural way the Borel algebra $\mathcal{B}(X)$, i.e., the smallest algebra (or $\sigma$-algebra) of subsets of $X$ containing all open subsets of $X$....
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1answer
271 views

Can someone explain Grothendieck schemes in layman language

Grothendieck Schemes are one of the key foundations of algebraic geometry which has been used in other fields as well Can anybody explain it in layman language? I just want the idea and not the ...
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Equivalence of two definitions of a sheaf

I don't understand why the definition of a sheaf (Definition 17.3.1 (ii)) given in the book [KS] Categories and Sheaves by Kashiwara and Schapira is equivalent to the definition of a sheaf (...
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An example in Mumford's “Picard Groups of Moduli Problems”

I'm reading Mumford's paper "Picard Groups of Moduli Problems" and am confused about an example in the first section. I'll try to explain the situation here, but if I'm not making sense I'm talking ...
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1answer
258 views

Stacks and Grothendieck topology.

I was reading https://math.dartmouth.edu/~jvoight/notes/moduli-red-harvard.pdf, introduction of some notes on algebraic geometry (stacks, I do not know what it is) and came across following statement ...
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Characterisation of $\mathbf{Hom}(C,F)$

$\def\C{\underline C}\def\hom{\mathbf{Hom}}\def\ker{\mathbf{Ker}}$ Let $\C$ be a category, let $C$ be a sieve on $\C$ with a basis $\{S_i\}$ and let $F:C^{op}\to\mathbf{Set}$ be a functor. Then ...
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1answer
121 views

Exactness of direct image functor of presheaves

Suppose $f:X\longrightarrow Y$ is a morphism of schemes. Take the categories $\mathbf{X}_{et},\,\mathbf{Y}_{et}$ of étale morphisms over $X$ and $Y$. Then is the direct image functor: $f_{*}:\mathbf{...
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1answer
253 views

Dense topology <=> double negation operator in a constructive metatheory?

The dense topology for a category $\mathbb{C}$ can be defined as follows, writing $\mathscr{H}:\mathbb{C}\to\widehat{\mathbb{C}}$ for the Yoneda embedding (considering sieves on $c$ as subfunctors of $...
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For which topologies is the pointwise coproduct already a sheaf?

In general, the coproduct of two sheaves is not the coproduct of their underlying presheaves. However, for some topologies, these notions do coincide (e.g. the chaotic topology). Can anyone point me ...
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1answer
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Schanuel topos: singleton coverage vs atomic topology

I have seen two different definitions of the Schanuel topos: Topos of sheaves on $\mathbf{FinSet}_{\mathsf{mono}}^{\mathsf{op}}$ with the atomic topology (all non-empty sieves cover) Topos of sheaves ...
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1answer
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Expressing “a subobject is a union of other subobjects” in a topos

Let $(\mathsf C,J)$ be a site. Let $U\rightarrowtail X$. Suppose there's a family of subobjects $(U_i\rightarrowtail X)$ which factors through $U\rightarrowtail X$ such that the factorized family $$(...
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1answer
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Characterization of covering families in terms of joint epimorphy of images

Suppose $\mathcal E\simeq \mathsf{Sh}(\mathsf C,J)$. I know the a family $(f_i:C_i\to C)_i$ covers $C$ iff the family $(\mathbf{ay}(c_i:C_i\to C))_i$ is jointly epimorphic in $\mathcal E$, where $\bf ...
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Proving $\operatorname{Spec} k[x]$ is internally a ring of fractions in the internal language of the Zariski topos

Let $\mathcal E$ be a Grothendieck topos and $R$ be a ring object in it. Say $R$ is internally a ring of fractions if $\neg (a=0)\implies a$ is invertible. I'm trying to prove that in the opposite ...
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Under-site like subspace topology

Let $X$ be a topological space and $Op(X)$ the category of its open sets. It is well known that $Op(X)$ has a canonical Grothendieck topology which makes of it a site. Let $U\in Op(X)$ be an object (...
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321 views

Sober Topological Space and its associated site

I was reading recently for Grothendieck topologies on an arbitrary category $\mathcal{C}$ (i.e. sites) and although did understand a couple of things I read something on Wiki article https://en....
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Why is the fundamental group a sheaf in the etale topology?

In this paper by Minhyong Kim on p5, there is a variety $X$ defined over $\mathbb{Q}$, $G = \pi_1(X(\mathbb{C}),b)$ the topological fundamental group of the associated complex algebraic variety, and $...
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381 views

Grothendieck topology and relation with usual topologies

Recently I stumbled upon the definition of $\textbf{Grothendieck}$ $\textbf{topologies}$ of a category $\mathcal{C}$. I do know that is one of the most interesting parts of the contemporary algebraic ...
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Do the effective descent morphisms w.r.t the codomain fibration hint at the “right topology”?

An intuitive approach to basic descent theory for me started with open covers $ \left\{ U_i \right\}$, replaced them with a singleton covering $\coprod _iU_i\rightarrow X$, and then generalized to ...
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Fibered categories, introduction or notes

I would like to learn about fibered categories, I know basic category theory, but not algebraic geometry. Is there a text, or lecture notes, which motivate the definitions from fields other than ...
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102 views

Existence of pullback of fiber bundles from abstract nonsense?

Let $\mathsf C$ be a superextensive site with products. A trivial fiber bundle is a bundle $\pi:E\rightarrow B$ which is isomorphic to $\pi_1:B\times F\rightarrow B$ in $\mathsf{C}/B$. Let $\mathcal ...
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1answer
121 views

Why does this exact sequence of sheaves imply the maps are $G$-equivariant?

I'm confused about something in Tamme's Introduction to Étale Cohomology (page 27). Let $G$ be a group and let $G$-$\mathsf{Set}$ denote the category of left $G$-sets with $G$-equivariant maps as ...
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Structure Sheaves of Rigid Analytic Spaces: What is the right Value Category?

Let $K$ be a complete non-archimedean field and let $X$ be an affinoid $K$-space. Then the structure sheaf $\mathcal O_X$ of $X$ is defined first on the "weak Grothendieck topology", ie on affinoid ...
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Interesting sites without pull-backs

I've seen at least a couple of sources (e.g. the nLab) that do not require the covering families in a coverage to be closed under pull-backs. The question is simple: are there any interesting ...
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Intersection condition for a Grothendieck topology

I am a bit confused about constructing a Grothendieck topology from a Grothendieck pretopology, largely because I have a discrepancy in definitions of the former. According to all of the questions I'...
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1answer
111 views

Moving from sheaves over spaces to sheaves over sites

The first example of a sheaf that I have consciously come across is the sheaf of continuous (real) functions on some topological space. The fact it is a sheaf is equivalent to the pasting lemma, which ...
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335 views

Is representability of Zariski sheaves local on the base?

Let $F: \mathsf{Sch_{/S}}^{op} \to \mathsf{Set}$ be a Zariski sheaf on the category of $S$-schemes. $F$ being a sheaf means it satisfies the following property: Sheaf condition: For every $S$-...
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Set-theoretical problems of regarding Grothendieck topologies as presheaves

I apologise for the vagueness of this question - it's just something I was idly wondering about. Let $(\mathcal{C},J)$ be a site with $\mathcal{C}$ small. The axioms for the Grothendieck topology $J$ ...
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Description of generated Grothendieck topology

Let $C$ be a small category, and let $\tau$ be a set of sieves in $C$. Assume that $\tau$ contains all the maximal sieves, and is stable under pullbacks. How to describe the Grothendieck topology $\...
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244 views

Covering sieves in a Grothendieck topology

I'm trying to get my head around some of the basics of Grothendieck topologies. Let $(\mathcal{C}, J)$ be a site, let $U$ be an object of $\mathcal{C}$ and let $J(U)$ be the set of covering sieves on ...