Sieve on an object - Imported Knowlpedia Content

Definition

Let $\mathcal C$ be a category and let $U$ be an object of $\mathcal C$ . A sieve on $U$ is a collection $S$ of morphisms with codomain $U$ that is closed under precomposition: if $f:V\to U$ belongs to $S$ and $g:W\to V$ is any morphism, then

f\circ g:W\longrightarrow U

also belongs to $S$ . Thus a sieve remembers a collection of ways to map into $U$ , together with every refinement of each such map.

Generated and maximal sieves

The maximal sieve on $U$ contains every morphism into $U$ . A family $\{f_i:U_i\to U\}$ generates the smallest sieve containing all the $f_i$ ; it consists of the morphisms into $U$ that factor through at least one $f_i$ .

Pullback

If $f:V\to U$ is a morphism, the pullback sieve $f^*S$ on $V$ consists of the morphisms $g:W\to V$ for which $f\circ g$ belongs to $S$ :

f^*S=\{g:W\to V\mid f\circ g\in S\}.

A Grothendieck topology specifies which sieves count as covering. Equivalently, a covering family is one whose generated sieve is covering.

Why sieves are useful

A covering family is a chosen list of local charts. Its generated sieve also includes every chart obtained by refining one of them, so it is independent of redundant choices in the original list. This makes sieves the natural categorical form of “all local probes compatible with this cover.”