## Sieve

### December 17th, 2006

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##

Sieve

**Definition 1.12**

*A*

**sieve**on an object is a collection S of morphisms in whose codomain is A and such that, if then, given any morphisms we have , i.e. S is closed under left composition:A map between sieves exists iff and given then

where

An important property of sieves is the following: if belongs to S which is a sieve on A, then the pullback of S by f determines a principal sieve on B, i.e.

An important properties of sieves is that the set of sieves defined on an object forms an Heyting algebra (see 3.5), with partial ordering given by subset inclusion.

Cecilia Flori 2007-01-02

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