The grothendieck-topology tag has no wiki summary.
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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 ...
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Which dense inclusions of sites are ∞-dense?
An inclusion of sites f: D→C is dense if it induces an equivalence between the categories of sheaves on C and D.
Likewise, f is ∞-dense is it induces an equivalence between the ∞-categories of ...
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353 views
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 ...
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Can one construct Freyd-Mitchell's embeddings that respect sheafifications?
For a certain presheaf $P$ with values in an abelian category $A$ satisfying AB5 and its sheafification $S$ (with respect to a small Grothendieck site) I would like to prove: $S(f):S(X)\to S(Y)$ is ...
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Can Inequivalent Topologies Have Same Sheaves/Cohomology?
Let $C$ be a fixed category, and let $T_1$ and $T_2$ be two Grothendieck (pre)topologies on $C$. We say $T_1$ is subordinate to $T_2$ if every covering in $T_1$ has a refinement in $T_2$. We say $T_1$ ...
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Needless axiom for Grothendieck topologies?
Hi,
The first axiom for a Grothendieck (pre)topology on a category $C$ says that for every object $X\in C$, the family consisting of just the identity $1_X : X\to X$ should be a covering family.
Why ...
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Is there a direct proof that affine schemes are fppf quasi-compact?
Let $A$ be a (commutative) ring. A family $(B_i)_{i\in I}$ of $A$-algebras is said to be an fppf cover if it satisfies three properties: (1) each $B_i$ is flat as an $A$-module, (2) each $B_i$ is ...
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cech cohomology in topos
Hi,
The following result seems to be well known, but I can't come up with a proof.
Suppose that $C$ is a topos and that $F\to G$ is an effective epimorphism in $C$. If $P$ is
any abelian sheaf on ...
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4answers
549 views
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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Examples of Sheafification via Hypercovers
For a presheaf $F$ on a category equipped with a pretopology, one has the sheafification $F^{\sharp}$ of $F$.
I know well the plus-construction of sheafification, which is presented in Artin's paper ...
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Numerable covers from the point of view of Grothendieck topologies
Let $G$ be a topological group. Recall that its classifying space $BG$ is a CW-complex which is the base of a locally trivial principal bundle of group $G$, with contractible total space $EG$. It ...
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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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Representable Presheaf
I have a very quick question. Is there an easy example of a representable presheaf on a site that is not a sheaf? This certainly can't happen on a small FPPF site so I would expect a counterexample to ...
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Grothendieck Topologies versus Pretopologies
The wikipedia article(s) as well as the nlab article(s) about Grothendieck topologies and Grothendieck pretopologies are careful to differentiate the two very emphatically and to point out that ...
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Coverage, itself considered as a presheaf
A coverage $J$ on a category $C$ assigns to an object $U$ of $C$ a set of covering families $J(U)$. The covering families are required to be stable under pullback, which amounts to requiring that for ...
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What is your picture of the flat topology?
I recently tried to explain the fppf site to a differential geometer. I started with the etale site, where I had two motivating claims:
If X is a smooth projective variety over the complexes, the ...
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Does this Grothendieck topology have a name?
I have the following Grothendieck pretopologies on the category of schemes.
The first one, the covers are families of morphisms $\{ U_i \to X \}$ such that for every point $x \in X$ there exists some ...
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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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1answer
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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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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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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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1answer
560 views
Commuting Grothendieck topologies.
Let $T_1$ and $T_2$ be two Grothendieck topologies on the same small category $C$, and let $T_3 = T_1 ∪ T_2$ (by which I mean the smallest Grothendieck topology on C containing $T_1$ and $T_2$).
...
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Nisnevich topology on non-(locally) Noetherian schemes
Background
Lurie has in DAG XI a definition (given below) of a Nisnevich cover for arbitrary commutative rings, which reduces to the usual one for Noetherian rings. It boils down to being a etale ...
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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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4answers
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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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392 views
Question about the definition of a sheaf cohomology group for a sheaf using tensor products of sheaves
In Warner's 'Foundations of differentiable manifolds and Lie groups', in the section about axiomatic sheaf theory (page 178), when establishing the conditions necessary for the existence of a ...
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What sheaf topoi classify: attribution request
Is there an accepted name or attribution by which to refer to the following well-known theorem?
If C is a small site, then the topos of sheaves on C is the classifying topos for flat ...
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1answer
665 views
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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Crystalline cohomology via the syntomic site
Hello,
Let $k$ be a field of characteristic $p > 0$, and let $Y$ be a $k$-scheme. Consider the
sites $Y_{syn}$ and $(Y/W_n)_{cris}$ (where $W_n$ are the Witt vectors of $k$ of length $n$), of $Y$ ...
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1answer
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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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4answers
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In what sense is the étale topology equivalent to the Euclidean topology?
I have heard it said more than once—on Wikipedia, for example—that the étale topology on the category of, say, smooth varieties over $\mathbb{C}$, is equivalent to the Euclidean topology. I have not ...
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Locally constant sheaves for the étale topology, lack of intuition about “étale-localness”
I have started studying some étale cohomology and I am trying to build up some intuition about the concept of local for the étale topology. I can understand some nice examples (like Kummer exact ...
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Stable motivic cohomology with finite coefficients?
In this question, which attracted no responces so far, I've asked whether it is possible to extend the Beilinson-Lichtenbaum etale descent rule for motivic cohomology to singular varieties, in ...
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1answer
625 views
Are completions stalks under some Grothendieck topology?
Let $R$ be a ring, and $\mathfrak{p}$ be a prime ideal. The stalk at $\mathfrak{p}$ with respect to the etale topology is $(R_{\mathfrak{p}})^{sh}$ (the strict henselization of $R_{\mathfrak{p}}$). ...
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Topologies (and sheaves) on Cat and CAT
I've been wondering lately what sort of Grothendieck (pre)topologies there are on $Cat$ (the category of small categories) and $CAT$ (the v. large category of large categories - to forestall criticism ...
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1answer
658 views
Motivic cohomology with finite coefficients for singular varieties
Let $X$ be a smooth variety over a field $K$ whose characteristic does not divide a positive integer $m$. Then the motivic cohomology of $X$ with coefficients in $\mathbb Z/m(j)$ can be computed in ...
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1answer
662 views
Zariski sheaves lifted to etale topology
Let $X$ be a "reasonable" scheme (I am particularly interested in smooth algebraic varieties over a field). Let $Zar_X$ denote the (small) Zariski site of (open subschemes of) $X$ and $Et_X$ denote ...
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Colimits of covers
Suppose I have category $C$ equipped with a Grothendiek pretopology of covers, and let $y:C \to Sh(C)$ be the Yoneda embedding into sheaves and $y/c:C/c \to Sh(C)/y(c)\cong Sh(C/c)$. How can I show ...
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1answer
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Grothendieck topologies, Mayer-Vietoris, and points
I am trying to think about certain problems in the theory of motives without having a proper background in Grothendieck topologies and the like, hoping to teach myself the related techniques in the ...
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Example of a Grothendieck pretopology satisfying a weak saturation condition
Recall that a singleton Grothendieck pretopology (henceforth 'singleton pretopology') on a category $C$ is a collection of maps $J$ containing the isomorphisms, closed under composition and stable ...
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1answer
336 views
Maps that admit local sections through each 'point' in the domain
In a recent MO question I asked about the relation between surjective submersions (in the category of smooth or otherwise manifolds) and maps that admit local sections. The latter, it turns out, are ...
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Is there a category of topological spaces such that open surjections admit local sections?
The class of open surjections $Q \to X$ is a Grothendieck pretopology on the category $Top$ of spaces, and includes the class of maps $\amalg U_\alpha \to X$ where $\{U_\alpha\}$ is an open cover of ...
