Site

topologized category.

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 $ {\mathcal O} ( X) $ of open sets of a topological space $ X $, 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 $ X $ is then just a functor from $ {\mathcal O} ( X) ^ { \mathop{\rm op} } $ to the category $ \mathbf{Ens} $ of sets; a pre-sheaf $ F $ is a sheaf if, for any covering of an open set $ U $ by smaller open sets $ U _ {i} $( $ i \in I $), the diagram

$$ F( U) \rightarrow \prod _ {i \in I } F( U _ {i} ) \begin{array}{c} \rightarrow \\ \rightarrow \end{array} \ \prod _ {( i,j) \in I \times I } F ( U _ {i} \cap U _ {j} ) $$

(where the arrows are induced in the obvious way by restriction mappings, i.e. by the action of $ F $ on morphisms of $ {\mathcal O} $) is an equalizer. (In more elementary terms, this says that elements of $ F( U) $ can be uniquely "patched together" from compatible families of elements of $ F( U _ {i} ) $.)

Abstracting from this definition, one defines a pre-sheaf on an arbitrary category $ C $ to be a functor $ C ^ { \mathop{\rm op} } \rightarrow \mathbf{Ens} $. In order to define the notion of a sheaf on $ C $, one needs a structure $ \tau $, called a Grothendieck topology, assigning to each object $ U $ of $ C $ a set $ \mathop{\rm Cov} ( U) $ of coverings of $ U $, which are families of morphisms $ ( f _ {i} : U _ {i} \rightarrow U) _ {i \in I } $ with common codomain $ U $. The assignment $ U \rightarrow \mathop{\rm Cov} ( U) $ is required to satisfy certain conditions, of which the most important is the "pullback-stability" condition:

a) If $ ( f _ {i} : U _ {i} \rightarrow U) _ {i \in I } \in \mathop{\rm Cov} ( U) $ and $ g: V \rightarrow U $ is any morphism, there exists $ ( h _ {j} : V _ {j} \rightarrow V) _ {j \in J } \in \mathop{\rm Cov} ( V) $ such that, for each $ j \in J $, the composite $ gh _ {j} : V _ {j} \rightarrow U $ factors through some $ f _ {i} $.

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 $ U $ of $ C $, the singleton family $ ( 1 _ {U} : U \rightarrow U ) $ is in $ \mathop{\rm Cov} ( U) $;

c) if $ ( f _ {i} : U _ {i} \rightarrow U) _ {i \in I } \in \mathop{\rm Cov} ( U) $ and, for each $ i $, $ ( g _ {ij} : U _ {ij} \rightarrow U _ {i} ) _ {j \in J _ {i} } \in \mathop{\rm Cov} ( U _ {i} ) $, then the family of all composites $ f _ {i} g _ {ij} : U _ {ij} \rightarrow U $( $ i \in I $, $ j \in J _ {i} $) is in $ \mathop{\rm Cov} ( U) $;

d) any family containing a family in $ \mathop{\rm Cov} ( U) $ is in $ \mathop{\rm Cov} ( U) $.

In defining the notion of a Grothendieck topology, many authors require the underlying category $ C $ to have pullbacks; in this case condition a) can be formulated more simply, but the restriction is not essential.

Given a Grothendieck topology $ \tau $ on $ C $, a pre-sheaf $ F $ on $ C $ is called a $ \tau $- sheaf (or simply a sheaf) if, for every family $ R =( f _ {i} : U _ {i} \rightarrow U) _ {i \in I } \in \mathop{\rm Cov} ( U) $, the canonical mapping $ F( U) \rightarrow F( R) $ induced by the $ F( f _ {i} ) $ is bijective, where $ F( R) $ denotes the set of families $ ( s _ {i} ) _ {i \in I } \in \prod _ {i \in I } F ( U _ {i} ) $ which are compatible in the sense that, whenever mappings $ g : V \rightarrow U _ {i} $ and $ h : V \rightarrow U _ {j} $ satisfy $ f _ {i} g = f _ {j} h $, then $ F( g)( s _ {i} )= F( h)( s _ {j} ) $. (Again, this definition can be stated more simply if the pullbacks $ U _ {i} \times _ {U} U _ {j} $ all exist in $ C $, but this is inessential.) Sheaves of Abelian groups, rings and other structures can be defined similarly.

The full subcategory of the functor category $ \widehat{C} = [ C ^ { \mathop{\rm op} } , \mathbf{Ens} ] $ whose objects are sheaves (for a given topology $ \tau $) is denoted by $ Sh ( C , \tau ) $ or $ \widetilde{C} $. Provided the site $ ( C, \tau ) $ satisfies an appropriate smallness condition, $ Sh ( C , \tau ) $ is a topos, and is a reflective subcategory of $ [ C ^ { \mathop{\rm op} } , \mathbf{Ens} ] $, the reflector preserving finite limits. Conversely, any reflective subcategory of $ [ C ^ { \mathop{\rm op} } , \mathbf{Ens} ] $ whose reflector preserves finite limits may be represented as $ Sh ( C, \tau ) $ for a suitable Grothendieck topology $ \tau $ on $ C $( Giraud's little theorem). Categories equivalent to one of the form $ Sh ( C, \tau ) $ are commonly called Grothendieck toposes (see Topos); they can be characterized (Giraud's big theorem) as categories $ E $ with the following properties:

1) $ E $ has finite limits;

2) $ E $ has arbitrary small coproducts, which are disjoint and universal (i.e. stable under pullback);

3) equivalence relations in $ E $ are effective, and have universal co-equalizers;

4) $ E $ has small $ \mathop{\rm hom} $- sets and a small set of generators.

Alternatively, a Grothendieck topos may be characterized as a category $ E $ 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 $ H ^ {i} ( C; F ) $, where $ F $ is a sheaf of Abelian groups on $ C $, are (the values at $ F $ of) the derived functors of the global section functor $ F \mapsto F( 1) $( where $ 1 $ is a terminal object of $ C $).

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.

Subsequently, they have been found useful in other contexts, notably in the construction of models for synthetic differential geometry [a3], [a4].

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