Sieve (category theory)
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In category theory, a branch of mathematics, a sieve is a way of choosing arrows with a common codomain. It is a categorical analogue of a collection of open subsets of a fixed open set in topology. In a Grothendieck topology, certain sieves become categorical analogues of open covers in topology. Sieves were introduced by Giraud (1964) in order to reformulate the notion of a Grothendieck topology.
Let C be a category, and let c be an object of C. A sieve on c is a subfunctor of Hom(−, c), i.e., for all objects c′ of C, S(c′) ⊆ Hom(c′, c), and for all arrows f:c″→c′, S(f) is the restriction of Hom(f, c), the pullback by f (in the sense of precomposition, not of fiber products), to S(c′); see the next section, below.
Put another way, a sieve is a collection S of arrows with a common codomain that satisfies the condition, "If g:c′→c is an arrow in S, and if f:c″→c′ is any other arrow in C, then gf is in S." Consequently, sieves are similar to right ideals in ring theory or filters in order theory.