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(→Definition: removed words "pullback" and "functoriality" from definition of sieve, because they are unnecessary  they mainly reduce the number of people who could understand it.) 

Let '''C''' be a [[Category (mathematics)category]], and let ''c'' be an object of '''C'''. A '''sieve''' <math>S\colon C^{\rm op} \to {\rm Set}</math> 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
==Pullback of sieves==
