Questions tagged [grothendieck-topology]
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125 questions
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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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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\...
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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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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 ...
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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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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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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 there a name for this "weak compatibility" between Grothendieck (pre)topologies?
(I find it easier to think in terms of Grothendieck pretopologies, instead of topologies. If this annoys any experts, please forgive me.)
Suppose that $C$ is a (full) subcategory of a category $D$. ...
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Induced pretopologies on sSet
Recall that the geometric realisation functor $| - |: sSet \to Top$ preserves products (choosing $Top = k Space$ or similar). Thus any given singleton Grothendieck pretopology on $Top$ gives rise to a ...
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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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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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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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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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Cohomology and quotients for the canonical topology
Recall that for any category $\mathcal C$, there is a unique finest topology, the canonical topology on $\mathcal C$ for which all representable functors are sheaves. I am interested in the example $\...
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What are the easiest cases of base change (for sheaves on sites)?
I have a closed embedding of schemes $i:X'\to X$, and for each of them I consider three Grothendieck topologies for the category of the corresponding (relatively) \'etale schemes: the \'etale one, the ...
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Fine and acyclic sheaves on locales
Hey all. Here's the thing, so I have a measure space. According to Johnstone's 'Topos theory' (page 213), let $(X,\Sigma,\mu)$ be a measure space, we can define a Grothendieck pretopology on it (and ...
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What was the original/historical motivation for introducing Grothendieck (pre-)topologies
The title essentially explains it, but I'll give some background:
I'm giving a talk to some fellow grad students about the relative Picard functor which requires introducing Grothendieck (pre-)...
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Do Categorical Quotients Preserve Covering Maps?
Before asking a question, please let me write down settings.
SETTINGS:
Let $C$ be a category with fiber products and $B$ be a closed subcategory of $C$ (i.e. $B$ contains any isomorphism of $C$, and ...
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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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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 ...
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Is any presheaf on a local ring a sheaf? Stalks/points for some Grothendieck topologies
I was able to recover the original statement, which is:
Let A be a local ring (for Zariski, henselian for Nisnevic and strictly henselian for étale) and $\{U_i\rightarrow Spec A\}$ a covering. Then ...
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Sheaffication using a $\lambda$-transfinite colimit
I asked this question on mathstack (long time ago), however I received no answers, so I'm trying it here. I don't know whether it's suitable for this site, anyway.
I was reading this article http://...
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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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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 ...
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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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