Questions tagged [grothendieck-topology]
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125 questions
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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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- 1,347
7
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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 ...
- 16.9k
2
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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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6
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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? ...
- 35.5k
4
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1
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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 cover-...
- 66.8k
6
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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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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 $Y$ ...
- 2,842
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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 ...
- 35.5k
37
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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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2
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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 ...
- 291
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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 ...
- 15.6k
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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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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 ...
- 15.6k
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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 ...
- 15.6k
4
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2
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695
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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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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 ...
- 15.6k
4
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0
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518
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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 ...
- 35.5k
4
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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 ...
- 35.5k
12
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Stacks in the fpqc topology
This is related to Matt Satriano's earlier question about an analog of Artin's theorem for stacks with an fpqc cover by a scheme.
Suppose one developed the theory of stacks in the fpqc topology and ...
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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 $...
- 35.5k
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Cohomology of sheaves in different Grothendieck topologies
Suppose I have a sheaf $\mathcal{F}$ on the (small) étale site over $X$. By restriction, $\mathcal{F}$ is also a sheaf on $X$ (with the Zariski topology). When is it that the sheaf cohomologies (i.e. ...
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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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