Sieve (category Theory) - Properties of Sieves

Properties of Sieves

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 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:

• 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).