A sieve over an object of category is the maximal sieve over if it contains all morphisms with target . In other words, it agrees with the class of all objects of the slice category . One of the axioms of Grothendieck topologies says that any maximal sieve (that is the maximal sieve for any object in ) is a covering sieve.
A maximal sieve is any sieve generated by an identity morphism in (recall that a sieve generated by a family of morphisms is the class of all morphisms of the form where is a morphism in ). If a sieve over is considered as a subpresheaf of the representable presheaf , then a sieve is maximal iff it is .