maximal sieve

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

A maximal sieve is any sieve generated by an identity morphism in CC (recall that a sieve generated by a family of morphisms {g i:c ic} iI\{g_i:c_i\to c\}_{i\in I} is the class of all morphisms of the form g ihg_i\circ h where h:ec ih:e\to c_i is a morphism in CC). If a sieve over cc is considered as a subpresheaf of the representable presheaf Hom(,c)Hom(-,c), then a sieve is maximal iff it is Hom(c)Hom(-c).

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