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Sieve (category theory) : ウィキペディア英語版 | Sieve (category theory) 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. ==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 is in ''S''." Consequently sieves are similar to right ideals in ring theory or filters in order theory.
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