A category equipped with a Grothendieck topology, that is, with a structure of "coverings" which makes it possible to define the notion of a sheaf on the category. The motivating example has as underlying category the lattice of open sets of a topological space , regarded as a category whose objects are the open sets and whose morphisms are the inclusion mappings between them. A pre-sheaf (of sets) on is then just a functor from to the category of sets; a pre-sheaf is a sheaf if, for any covering of an open set by smaller open sets (), the diagram
(where the arrows are induced in the obvious way by restriction mappings, i.e. by the action of on morphisms of ) is an equalizer. (In more elementary terms, this says that elements of can be uniquely "patched together" from compatible families of elements of .)
Abstracting from this definition, one defines a pre-sheaf on an arbitrary category to be a functor . In order to define the notion of a sheaf on , one needs a structure , called a Grothendieck topology, assigning to each object of a set of coverings of , which are families of morphisms with common codomain . The assignment is required to satisfy certain conditions, of which the most important is the "pullback-stability" condition:
a) If and is any morphism, there exists such that, for each , the composite factors through some .
Other closure conditions which are commonly imposed, though they are less important for the purpose of defining the category of sheaves, are:
b) for every object of , the singleton family is in ;
c) if and, for each , , then the family of all composites (, ) is in ;
d) any family containing a family in is in .
In defining the notion of a Grothendieck topology, many authors require the underlying category to have pullbacks; in this case condition a) can be formulated more simply, but the restriction is not essential.
Given a Grothendieck topology on , a pre-sheaf on is called a -sheaf (or simply a sheaf) if, for every family , the canonical mapping induced by the is bijective, where denotes the set of families which are compatible in the sense that, whenever mappings and satisfy , then . (Again, this definition can be stated more simply if the pullbacks all exist in , but this is inessential.) Sheaves of Abelian groups, rings and other structures can be defined similarly.
The full subcategory of the functor category whose objects are sheaves (for a given topology ) is denoted by or . Provided the site satisfies an appropriate smallness condition, is a topos, and is a reflective subcategory of , the reflector preserving finite limits. Conversely, any reflective subcategory of whose reflector preserves finite limits may be represented as for a suitable Grothendieck topology on (Giraud's little theorem). Categories equivalent to one of the form are commonly called Grothendieck toposes (see Topos); they can be characterized (Giraud's big theorem) as categories with the following properties:
1) has finite limits;
2) has arbitrary small coproducts, which are disjoint and universal (i.e. stable under pullback);
3) equivalence relations in are effective, and have universal co-equalizers;
4) has small -sets and a small set of generators.
Alternatively, a Grothendieck topos may be characterized as a category with a set of generators, which is equivalent to the category of sheaves on itself when it is equipped with the canonical topology (the largest topology for which all representable functors are sheaves, cf. Representable functor).
The category of Abelian groups in a Grothendieck topos (equivalently, the category of sheaves of Abelian groups on a site) is a Grothendieck category, which makes it possible to define sheaf cohomology on a site; the cohomology groups , where is a sheaf of Abelian groups on , are (the values at of) the derived functors of the global section functor (where is a terminal object of ).
Sites were first introduced in algebraic geometry [a1], [a2], in connection with the étale topology of a scheme and similar topologies used to define cohomology theories studied by algebraic geometers.
|[a1]||J. Giraud, "Analysis situs" , Sem. Bourbaki (1963) pp. Exp. 256 MR1611542 MR0193122 Zbl 0201.23302|
|[a2]||M. Artin, A. Grothendieck, J.-L. Verdier, "Théorie des topos et cohomologie étale des schémas" , SGA 4 , Lect. notes in math. , 269; 270; 305 , Springer (1972) MR0354653 MR0354652|
|[a3]||A. Kock, "Synthetic differential geometry" , Cambridge Univ. Press (1981) MR0649622 MR0596153 Zbl 0487.18006 Zbl 0466.51008|
|[a4]||I. Moerdijk, G.E. Reyes, "Models for smooth infinitesimal analysis" , Springer (1990) MR1083355 Zbl 0715.18001|
|[a5]||P.T. Johnstone, "Topos theory" , Acad. Press (1977) MR0470019 Zbl 0368.18001|
Site. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Site&oldid=42570