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A category equipped with a Grothendieck topology.
A site is a pair , where is a category and is a Grothendieck topology on .
The topology determines which sieves, or equivalently which covering families, may be used to formulate local definitions and gluing conditions. A site is categorical data: its objects are not required to be open subsets of one topological space.
A category consists of:
These data satisfy the category axioms (associativity and identity/unit laws); see category axioms.
This abstracts the behavior of functions and their composition.
Let be a category. A sieve on an object is a collection of morphisms with codomain that is closed under precomposition: if belongs to , then every composite belongs to .
A Grothendieck topology on assigns to every object a collection of sieves, called covering sieves, such that:
Thus a Grothendieck topology specifies what it means to work locally on the objects of a category; it need not arise from a topology on a set of points.
Let be a site and let be an object of . A family of morphisms
is a covering family if the sieve it generates belongs to . The generated sieve consists of all morphisms that factor through at least one .
This definition depends on the chosen Grothendieck topology. For the étale topology, the covering families are precisely the jointly surjective families of étale morphisms.