In category theory, a branch of mathematics, a sieve is a way of choosing arrows with a common codomain.
Sieves were introduced by Giraud (1964) in order to reformulate the notion of a Grothendieck topology.
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.
A Grothendieck topology is a collection of sieves subject to certain properties.
J(c) satisfies several properties in addition to those required by the definition: Consequently, J(c) is also a distributive lattice, and it is cofinal in Sieve(c).