site (changes) in nLab
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Context
Topos Theory
Background
Toposes
Internal Logic
Topos morphisms
Cohomology and homotopy
In higher category theory
Theorems
Contents
Idea
A site is a presentation of a sheaf topos as a structure freely generated under colimits from a category, subject to the relation that certain covering colimits are preserved.
As such, sites generalise topological spaces and locales, which present localic sheaf toposes. More precisely, sites generalise and categorify posites, which present localic toposes but also present locales themselves in a decategorified manner.
In technical terms, a site is a small category equipped with a coverage or Grothendieck topology. The category of sheaves over a site is a sheaf topos and the site is a site of definition for this topos.
Definition
Definition
A site (C,J)(C,J) is a category CC equipped with a coverage JJ.
For ℰ\mathcal{E} a topos equipped with an equivalence of categories
ℰ≃Sh(C,J) \mathcal{E} \simeq Sh(C,J)
to the sheaf topos over a site, one says that (C,J)(C,J) is a site of definition for ℰ\mathcal{E}.
Some classes of sites have their special names
Definition
A site is called
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a small site, large site, essentially small site if the underlying category is a small category, large category, essentially small category, respectively;
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a cartesian site if the underlying category is finitely complete (which the Elephant calls a cartesian category);
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a standard site if it is a cartesian site equipped with a subcanonical coverage.
The term standard site appears in (Johnstone, example A2.1.11).
Properties
Morita equivalent sites
Many inequivalent sites may have equivalent sheaf toposes.
This appears as (Johnstone, theorem C2.2.8 (iii)).
Subcanonical sites
This appears as (Johnstone, prop. C2.2.16).
Examples
Classes of sites
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Every frame is canonically a site, where UU is covered by {U i}\{U_i\} precisely if it is their join.
A subclass of examples is the category of open subsets of a topological space.
This are examples of posites/(0,1)-site.
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Various categories come with canonical structures of sites on them:
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For every category CC there is its canonical coverage.
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On every regular category there is its regular coverage.
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On every coherent category there is its coherent coverage.
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Generalizing the previous two examples, on an ∞-ary regular category there is a κ\kappa-canonical coverage.
If the category in question is the syntactic category of a theory, the corresponding site is the syntactic site.
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For every site there is the corresponding double negation topology that forces the sheaf topos to a Boolean topos.
Other classes of sites are listed in the following.
Specific sites
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Sites for big toposes defining notions of geometry are:
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The sites that define the geometry called differential geometry are CartSp smooth{}_{smooth}, SmoothMfd, etc, equipped with the open cover coverage. Or more generally smooth loci etc.
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The sites that induce topological geometry? are small versions of Top equipped with the open cover coverage.
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The sites that induce the higher geometry modeled on Euclidean topology are the large site of paracompact manifolds and its dense sub-site CartSp top{}_{top}.
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The sites that define the geometry called algebraic geometry are site structures on categories of formal duals of commutative rings or commutative associative algebras
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fpqc-site →\to fppf-site →\to syntomic site →\to étale site →\to Nisnevich site →\to Zariski site
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site
References
In
sites are discussed in section C2.1.
Last revised on August 4, 2016 at 07:27:42. See the history of this page for a list of all contributions to it.