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''topologized category.''

A category equipped with a [[Grothendieck topology|Grothendieck topology]], that is, with a structure of "coverings" which makes it possible to define the notion of a [[Sheaf|sheaf]] on the category. The motivating example has as underlying category the lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s0856601.png" /> of open sets of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s0856602.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s0856603.png" /> is then just a functor from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s0856604.png" /> to the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s0856605.png" /> of sets; a pre-sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s0856606.png" /> is a sheaf if, for any covering of an open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s0856607.png" /> by smaller open sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s0856608.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s0856609.png" />), the diagram

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566010.png" /></td> </tr></table>

(where the arrows are induced in the obvious way by restriction mappings, i.e. by the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566011.png" /> on morphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566012.png" />) is an equalizer. (In more elementary terms, this says that elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566013.png" /> can be uniquely "patched together" from compatible families of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566014.png" />.)

Abstracting from this definition, one defines a pre-sheaf on an arbitrary category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566015.png" /> to be a functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566016.png" />. In order to define the notion of a sheaf on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566017.png" />, one needs a structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566018.png" />, called a Grothendieck topology, assigning to each object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566019.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566020.png" /> a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566021.png" /> of coverings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566022.png" />, which are families of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566023.png" /> with common codomain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566024.png" />. The assignment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566025.png" /> is required to satisfy certain conditions, of which the most important is the "pullback-stability" condition:

a) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566027.png" /> is any morphism, there exists <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566028.png" /> such that, for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566029.png" />, the composite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566030.png" /> factors through some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566031.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566032.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566033.png" />, the singleton family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566034.png" /> is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566035.png" />;

c) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566036.png" /> and, for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566038.png" />, then the family of all composites <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566039.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566041.png" />) is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566042.png" />;

d) any family containing a family in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566043.png" /> is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566044.png" />.

In defining the notion of a Grothendieck topology, many authors require the underlying category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566045.png" /> to have pullbacks; in this case condition a) can be formulated more simply, but the restriction is not essential.

Given a Grothendieck topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566046.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566047.png" />, a pre-sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566048.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566049.png" /> is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566051.png" />-sheaf (or simply a sheaf) if, for every family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566052.png" />, the canonical mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566053.png" /> induced by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566054.png" /> is bijective, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566055.png" /> denotes the set of families <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566056.png" /> which are compatible in the sense that, whenever mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566058.png" /> satisfy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566059.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566060.png" />. (Again, this definition can be stated more simply if the pullbacks <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566061.png" /> all exist in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566062.png" />, but this is inessential.) Sheaves of Abelian groups, rings and other structures can be defined similarly.

The full subcategory of the [[functor category]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566063.png" /> whose objects are sheaves (for a given topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566064.png" />) is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566065.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566066.png" />. Provided the site <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566067.png" /> satisfies an appropriate smallness condition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566068.png" /> is a [[Topos|topos]], and is a [[Reflective subcategory|reflective subcategory]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566069.png" />, the reflector preserving finite limits. Conversely, any reflective subcategory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566070.png" /> whose reflector preserves finite limits may be represented as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566071.png" /> for a suitable Grothendieck topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566072.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566073.png" /> (Giraud's little theorem). Categories equivalent to one of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566074.png" /> are commonly called Grothendieck toposes (see [[Topos|Topos]]); they can be characterized (Giraud's big theorem) as categories <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566075.png" /> with the following properties:

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566076.png" /> has finite limits;

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566077.png" /> has arbitrary small coproducts, which are disjoint and universal (i.e. stable under pullback);

3) equivalence relations in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566078.png" /> are effective, and have universal co-equalizers;

4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566079.png" /> has small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566080.png" />-sets and a small set of generators.

Alternatively, a Grothendieck topos may be characterized as a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566081.png" /> 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|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|Grothendieck category]], which makes it possible to define sheaf cohomology on a site; the cohomology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566082.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566083.png" /> is a sheaf of Abelian groups on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566084.png" />, are (the values at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566085.png" /> of) the derived functors of the global section functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566086.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566087.png" /> is a terminal object of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085660/s08566088.png" />).

Sites were first introduced in algebraic geometry [[#References|[a1]]], [[#References|[a2]]], in connection with the [[Etale topology|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|synthetic differential geometry]] [[#References|[a3]]], [[#References|[a4]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Giraud, "Analysis situs" , ''Sem. Bourbaki'' (1963) pp. Exp. 256 {{MR|1611542}} {{MR|0193122}} {{ZBL|0201.23302}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> 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) {{MR|0354653}} {{MR|0354652}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A. Kock, "Synthetic differential geometry" , Cambridge Univ. Press (1981) {{MR|0649622}} {{MR|0596153}} {{ZBL|0487.18006}} {{ZBL|0466.51008}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> I. Moerdijk, G.E. Reyes, "Models for smooth infinitesimal analysis" , Springer (1990) {{MR|1083355}} {{ZBL|0715.18001}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> P.T. Johnstone, "Topos theory" , Acad. Press (1977) {{MR|0470019}} {{ZBL|0368.18001}} </TD></TR></table>