Sieve (category theory): Difference between revisions
no edit summary
Tags: Mobile edit Mobile web edit 

In [[category theory]], a branch of [[mathematics]], a '''sieve''' is a way of choosing
==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(−, ''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
==Pullback of sieves==
The most common operation on a sieve is ''pullback''. Pulling back a sieve ''S'' on ''c'' by an arrow ''f'':''c''′→''c'' gives a new sieve ''f''<sup>*</sup>''S'' on ''c''′. This new sieve consists of all the arrows in ''S''
There are several equivalent ways of defining ''f''<sup>*</sup>''S''. The simplest is:
Let ''S'' and ''S''′ be two sieves on ''c''. We say that ''S'' ⊆ ''S''′ if for all objects ''c''′ of '''C''', ''S''(''c''′) ⊆ ''S''′(''c''′). For all objects ''d'' of '''C''', we define (''S'' ∪ ''S''′)(''d'') to be ''S''(''d'') ∪ ''S''′(''d'') and (''S'' ∩ ''S''′)(''d'') to be ''S''(''d'') ∩ ''S''′(''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 [[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:
}}
*{{citationmr=0193122
last=Giraudfirst= Jeanauthorlink = Jean Giraud
chapter=Analysis situsyear=1964 title=Séminaire Bourbaki, 1962/63. Fasc. 3, issue= 256 publisher= Secrétariat mathématiqueplace= Paris url=http://www.numdam.org/item?id=SB_19621964__8__189_0}}
* {{cite book  editor1last=Pedicchio  editor1first=Maria Cristina  editor2last=Tholen  editor2first=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=0521834147  zbl=1034.18001 }}
