Sieve (category theory)
Encyclopedia
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
.
, and let c be an object of C. A sieve S 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′).
Put another way, a sieve is a collection S of arrows with a common codomain which satisfies the functoriality condition, "If g:c′→c is an arrow in S, and if f:c″→c′ is any other arrow in C, then the pullback is in S." Consequently sieves are similar to right ideal
s in ring theory
or filter
s in order theory
.
There are several equivalent ways of defining f*S. The simplest is:
A more abstract formulation is:
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′).
If we define SieveC(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:
Consequently, J(c) is also a distributive lattice
, and it is cofinal
in Sieve(c).
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a sieve is a way of choosing arrows with a common codomain
Codomain
In mathematics, the codomain or target set of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation...
. It is a categorical analogue of a collection of open subsets of a fixed open set
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
in topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
. In a Grothendieck topology
Grothendieck topology
In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site.Grothendieck topologies axiomatize the notion...
, certain sieves become categorical analogues of open covers in topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
.
Definition
Let C be a categoryCategory (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
, and let c be an object of C. A sieve S on c is a subfunctor
Subfunctor
In category theory, a branch of mathematics, a subfunctor is a special type of functor which is an analogue of a subset.-Definition:Let C be a category, and let F be a functor from C to the category of sets Set...
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
Pullback
Suppose that φ:M→ N is a smooth map between smooth manifolds M and N; then there is an associated linear map from the space of 1-forms on N to the space of 1-forms on M. This linear map is known as the pullback , and is frequently denoted by φ*...
by f (in the sense of precomposition, not of fiber products), to S(c′).
Put another way, a sieve is a collection S of arrows with a common codomain which satisfies the functoriality condition, "If g:c′→c is an arrow in S, and if f:c″→c′ is any other arrow in C, then the pullback is in S." Consequently sieves are similar to right ideal
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
s in ring theory
Ring theory
In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
or filter
Filter (mathematics)
In mathematics, a filter is a special subset of a partially ordered set. A frequently used special case is the situation that the ordered set under consideration is just the power set of some set, ordered by set inclusion. Filters appear in order and lattice theory, but can also be found in...
s in order theory
Order theory
Order theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and gives some basic definitions...
.
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*S on c′. This new sieve consists of all the arrows in S which 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 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 SieveC(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
Complete lattice
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum and an infimum . Complete lattices appear in many applications in mathematics and computer science...
.
A Grothendieck topology
Grothendieck topology
In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site.Grothendieck topologies axiomatize the notion...
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
Distributive lattice
In mathematics, distributive lattices are lattices for which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection...
, and it is cofinal
Cofinal (mathematics)
In mathematics, let A be a set and let ≤ be a binary relation on A. Then a subset B of A is said to be cofinal if it satisfies the following condition:This definition is most commonly applied when A is a partially ordered set or directed set under the relation ≤. Also, the notion of cofinal...
in Sieve(c).