# Sieve (category theory)

In category theory, a branch of mathematics, a **sieve** is a way of choosing arrows with a common codomain. It is a categorical analogue of a collection of open subsets of a fixed open set in topology. In a Grothendieck topology, certain sieves become categorical analogues of open covers in topology. Sieves were introduced by Giraud (1964) in order to reformulate the notion of a Grothendieck topology.

## DefinitionEdit

Let **C** be a category, and let *c* be an object of **C**. A **sieve** 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 condition, "If *g*:*c*′→*c* is an arrow in *S*, and if *f*:*c*″→*c*′ is any other arrow in **C**, then *gf* is in *S*." Consequently, sieves are similar to right ideals in ring theory or filters in order theory.

## Pullback of sievesEdit

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*^{*}*S* on *c*′. This new sieve consists of all the arrows in *S* that factor through *c*′.

There are several equivalent ways of defining *f*^{*}*S*. The simplest is:

- For any object
*d*of**C**,*f*^{*}*S*(*d*) = {*g*:*d*→*c*′ | fg ∈*S*(*d*)}

A more abstract formulation is:

*f*^{*}*S*is the image of the fibered product*S*×_{Hom(−, c)}Hom(−,*c*′) under the natural projection*S*×_{Hom(−, c)}Hom(−,*c*′)→Hom(−,*c*′).

Here the map Hom(−, *c*′)→Hom(−, *c*) is Hom(*f*, *c*′), the pullback by *f*.

The latter formulation suggests that we can also take the image of *S*×_{Hom(−, c)}Hom(−, *c*′) under the natural map to Hom(−, *c*). This will be the image of *f*^{*}*S* under composition with *f*. For each object *d* of **C**, this sieve will consist of all arrows *fg*, where *g*:*d*→*c*′ is an arrow of *f*^{*}*S*(*d*). In other words, it consists of all arrows in *S* that can be factored through *f*.

If we denote by ∅_{c} the empty sieve on *c*, that is, the sieve for which ∅(*d*) is always the empty set, then for any *f*:*c*′→*c*, *f*^{*}∅_{c} is ∅_{c′}. Furthermore, *f*^{*}Hom(−, *c*) = Hom(−, *c*′).

## Properties of sievesEdit

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_{C}(*c*) (or Sieve(*c*) for short) to be the set of all sieves on *c*, then Sieve(*c*) becomes 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:

- If
*S*and*S*′ are sieves on*c*,*S*⊆*S*′, and*S*∈*J*(*c*), then*S*′ ∈*J*(*c*). - Finite intersections of elements of
*J*(*c*) are in*J*(*c*).

Consequently, *J*(*c*) is also a distributive lattice, and it is cofinal in Sieve(*c*).

## ReferencesEdit

- Artin, Michael; Alexandre Grothendieck; Jean-Louis Verdier, eds. (1972).
*Séminaire de Géométrie Algébrique du Bois Marie - 1963-64 - Théorie des topos et cohomologie étale des schémas - (SGA 4) - vol. 1*. Lecture notes in mathematics (in French).**269**. Berlin; New York: Springer-Verlag. xix+525. doi:10.1007/BFb0081551. ISBN 978-3-540-05896-0. - Giraud, Jean (1964), "Analysis situs",
*Séminaire Bourbaki, 1962/63. Fasc. 3*, Paris: Secrétariat mathématique, MR 0193122 - Pedicchio, Maria Cristina; Tholen, Walter, eds. (2004).
*Categorical foundations. Special topics in order, topology, algebra, and sheaf theory*. Encyclopedia of Mathematics and Its Applications.**97**. Cambridge: Cambridge University Press. ISBN 0-521-83414-7. Zbl 1034.18001.