Sieve (category theory): Difference between revisions

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In [[category theory]], a branch of [[mathematics]], a '''sieve''' is a way of choosing arrows[[morphism|arrow]]s with a common [[codomain]]. It is a categorical analogue of a collection of open subsets[[subset]]s of a fixed [[open set]] in [[topology]]. In a [[Grothendieck topology]], certain sieves become categorical analogues of [[open cover]]s in [[topology]]. Sieves were introduced by {{harvtxt|Giraud|1964}} in order to reformulate the notion of a Grothendieck topology.
 
==Definition==
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 whichthat 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==
The most common operation on a sieve is ''pullback''. Pulling back a sieve ''S'' on ''c'' by an arrow ''f'':''c''&prime;→''c'' gives a new sieve ''f''<sup>*</sup>''S'' on ''c''&prime;. This new sieve consists of all the arrows in ''S'' whichthat factor through ''c''&prime;.
 
There are several equivalent ways of defining ''f''<sup>*</sup>''S''. The simplest is:
Let ''S'' and ''S''&prime; be two sieves on ''c''. We say that ''S'' ⊆ ''S''&prime; if for all objects ''c''&prime; of '''C''', ''S''(''c''&prime;) ⊆ ''S''&prime;(''c''&prime;). For all objects ''d'' of '''C''', we define (''S'' ∪ ''S''&prime;)(''d'') to be ''S''(''d'') ∪ ''S''&prime;(''d'') and (''S'' ∩ ''S''&prime;)(''d'') to be ''S''(''d'') ∩ ''S''&prime;(''d''). We can clearly extend this definition to infinite unions and intersections as well.
 
If we define Sieve<sub>'''C'''</sub>(''c'') (or Sieve(''c'') for short) to be the set of all sieves on ''c'', then Sieve(''c'') becomes a partially ordered under ⊆. It is easy to see from the definition that the union or intersection of any family of sieves on ''c'' is a sieve on ''c'', so Sieve(''c'') is a [[complete lattice]].
 
A [[Grothendieck topology]] is a collection of sieves subject to certain properties. These sieves are called ''covering sieves''. The set of all covering sieves on an object ''c'' is a subset ''J''(''c'') of Sieve(''c''). ''J''(''c'') satisfies several properties in addition to those required by the definition:
}}
*{{citation|mr=0193122
|last=Giraud|first= Jean|authorlink = Jean Giraud
|chapter=Analysis situs|year=1964 |title=Séminaire Bourbaki, 1962/63. Fasc. 3, |issue= 256 |publisher= Secrétariat mathématique|place= Paris |url=http://www.numdam.org/item?id=SB_1962-1964__8__189_0}}
* {{cite book | editor1-last=Pedicchio | editor1-first=Maria Cristina | editor2-last=Tholen | editor2-first=Walter | title=Categorical foundations. Special topics in order, topology, algebra, and sheaf theory | series=Encyclopedia of Mathematics and Its Applications | volume=97 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2004 | isbn=0-521-83414-7 | zbl=1034.18001 }}