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 $\{{g}_{i}:{c}_{i}\to c{\}}_{i\in I}$ is the class of all morphisms of the form ${g}_{i}\circ h$ where $h:e\to {c}_{i}$ is a morphism in $C$). If a sieve over $c$ is considered as a subpresheaf of the representable presheaf $\mathrm{Hom}(-,c)$, then a sieve is maximal iff it is $\mathrm{Hom}(-c)$.

Revised on July 11, 2009 05:28:25
by Toby Bartels
(71.104.230.172)