maximal sieve

A sieve over an object c of category C is the maximal sieve over c if it contains all morphisms with target c. In other words, it agrees with the class of all objects of the slice category C/c. One of the axioms of Grothendieck topologies says that any maximal sieve (that is the maximal sieve for any object in C) is a covering sieve.

A maximal sieve is any sieve generated by an identity morphism in C (recall that a sieve generated by a family of morphisms {g i:c ic} iI is the class of all morphisms of the form g ih where h:ec i is a morphism in C). If a sieve over c is considered as a subpresheaf of the representable presheaf Hom(,c), then a sieve is maximal iff it is Hom(c).

Revised on July 11, 2009 05:28:25 by Toby Bartels (