Definition
Let C be a category, and let c be an object of C. A sieve S 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′).
Put another way, a sieve is a collection S of arrows with a common codomain which satisfies the functoriality condition, "If g:c′→c is an arrow in S, and if f:c″→c′ is any other arrow in C, then the pullback S(f)(g) = gf is in S." Consequently sieves are similar to right ideals in ring theory or filters in order theory.
Read more about this topic: Sieve (category Theory)
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