Sieve (category theory): Difference between revisions

→‎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.
(→‎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(&minus;, ''c''), i.e., for all objects ''c''&prime; of '''C''', ''S''(''c''&prime;) ⊆ Hom(''c''&prime;, ''c''), and for all arrows ''f'':''c''&Prime;→''c''&prime;, ''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''&prime;); see the next section, below.
 
Put another way, a sieve is a collection ''S'' of arrows with a common codomain that satisfies the functoriality condition, "If ''g'':''c''&prime;&rarr;''c'' is an arrow in ''S'', and if ''f'':''c''&Prime;&rarr;''c''&prime; is any other arrow in '''C''', then the pullback {{nowrap|''S''(''f'')(''g'') {{=}} ''gf''}} is in ''S''." Consequently, sieves are similar to right [[Ideal (ring theory)|ideal]]s in [[ring theory]] or [[filter (mathematics)|filter]]s in [[order theory]].
 
==Pullback of sieves==