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.

Definition[edit]

Let C be a category, and let c be an object of C. A sieve $S\colon C^{\rm {op}}\to {\rm {Set}}$ 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 sieves[edit]

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), the push forward 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 sieves[edit]

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).

References[edit]

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). Vol. 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. Vol. 97. Cambridge: Cambridge University Press. ISBN 0-521-83414-7. Zbl 1034.18001.