The grothendieck-topology tag has no wiki summary.
5
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0answers
91 views
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 ...
1
vote
1answer
348 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 ...
3
votes
0answers
160 views
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 ...
5
votes
3answers
571 views
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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0answers
188 views
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 ...
9
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1answer
349 views
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 ...
2
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0answers
130 views
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 ...
7
votes
4answers
541 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 ...
1
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0answers
113 views
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 ...
2
votes
1answer
211 views
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 ...
1
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1answer
138 views
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 $D$. ...
7
votes
3answers
454 views
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 ...
8
votes
4answers
1k views
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 ...
3
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1answer
178 views
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 ...
26
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1answer
927 views
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 ...
5
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0answers
333 views
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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0answers
136 views
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 ...
1
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1answer
231 views
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 ...
15
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1answer
588 views
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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0answers
390 views
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 ...
6
votes
1answer
559 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 $T_2$).
...
5
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1answer
481 views
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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0answers
298 views
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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0answers
342 views
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 ...
31
votes
4answers
2k views
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? ...
1
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1answer
391 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 ...
7
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1answer
328 views
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 ...
3
votes
1answer
656 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.
...
14
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1answer
994 views
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 $Y$ ...
2
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1answer
221 views
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 ...
17
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4answers
2k views
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 ...
10
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2answers
1k views
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 ...
7
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0answers
309 views
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 ...
17
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1answer
619 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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0answers
216 views
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 ...
11
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1answer
655 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
658 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 ...
3
votes
2answers
297 views
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 ...
13
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1answer
937 views
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 ...
4
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0answers
265 views
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 ...
2
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1answer
334 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 ...
15
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0answers
460 views
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 ...
