Subfunctor
Encyclopedia
In category theory
, a branch of mathematics
, a subfunctor is a special type of functor
which is an analogue of a subset.
Set. A functor G from C to Set is a subfunctor of F if
This relation is often written as G ⊆ F.
For example, let 1 be the category with a single object and a single arrow. A functor F:1→Set maps the unique object of 1 to some set S and the unique identity arrow of 1 to the identity function 1S on S. A subfunctor G of F maps the unique object of 1 to a subset T of S and maps the unique identity arrow to the identity function 1T on T. Notice that 1T is the restriction of 1S to T. Consequently, subfunctors of F correspond to subsets of S.
The most important examples of subfunctors are subfunctors of the Hom functor. Let c be an object of the category C, and consider the functor Hom(−, c). This functor takes an object c′ of C and gives back all of the morphisms c′→c. A subfunctor of Hom(−, c) gives back only some of the morphisms. Such a subfunctor is called a sieve
, and it is usually used when defining Grothendieck topologies.
s on the category of ringed space
s. Let F be a functor from the category of ringed spaces to the category of sets, and let G ⊆ F. Suppose that this inclusion morphism G→F is representable by open immersions, i.e., for any representable functor Hom(−, X) and any morphism Hom(−, X)→F, the fibered product G×FHom(−, X) is a representable functor Hom(−, Y) and the morphism Y→X defined by the Yoneda lemma
is an open immersion. Then G is called an open subfunctor of F. If F is covered by representable open subfunctors, then, under certain conditions, it can be shown that F is representable. This is a useful technique for the construction of ringed spaces. It was discovered and exploited heavily by Alexandre Grothendieck, who applied it especially to the case of scheme
s. For a formal statement and proof, see Grothendieck, Elements de Geometrie Algebrique, vol. 1, 2nd ed., chapter 0, section 4.5.
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 subfunctor is a special type of functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
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 setsCategory of sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are all functions from A to B...
Set. A functor G from C to Set is a subfunctor of F if
- For all objects c of C, G(c) ⊆ F(c), and
- For all arrows f:c′→c of C, G(f) is the restriction of F(f) to G(c′).
This relation is often written as G ⊆ F.
For example, let 1 be the category with a single object and a single arrow. A functor F:1→Set maps the unique object of 1 to some set S and the unique identity arrow of 1 to the identity function 1S on S. A subfunctor G of F maps the unique object of 1 to a subset T of S and maps the unique identity arrow to the identity function 1T on T. Notice that 1T is the restriction of 1S to T. Consequently, subfunctors of F correspond to subsets of S.
Remarks
Subfunctors in general are like global versions of subsets. For example, if one imagines the objects of some category C to be analogous to the open sets of a topological space, then a functor from C to the category of sets gives a set-valued presheaf on C, that is, it associates sets to the objects of C in a way which is compatible with the arrows of C. A subfunctor then associates a subset to each set, again in a compatible way.The most important examples of subfunctors are subfunctors of the Hom functor. Let c be an object of the category C, and consider the functor Hom(−, c). This functor takes an object c′ of C and gives back all of the morphisms c′→c. A subfunctor of Hom(−, c) gives back only some of the morphisms. Such a subfunctor is called a sieve
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...
, and it is usually used when defining Grothendieck topologies.
Open subfunctors
Subfunctors are also used in the construction of representable functorRepresentable functor
In mathematics, particularly category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures In mathematics, particularly category theory, a...
s on the category of ringed space
Ringed space
In mathematics, a ringed space is, intuitively speaking, a space together with a collection of commutative rings, the elements of which are "functions" on each open set of the space...
s. Let F be a functor from the category of ringed spaces to the category of sets, and let G ⊆ F. Suppose that this inclusion morphism G→F is representable by open immersions, i.e., for any representable functor Hom(−, X) and any morphism Hom(−, X)→F, the fibered product G×FHom(−, X) is a representable functor Hom(−, Y) and the morphism Y→X defined by the Yoneda lemma
Yoneda lemma
In mathematics, specifically in category theory, the Yoneda lemma is an abstract result on functors of the type morphisms into a fixed object. It is a vast generalisation of Cayley's theorem from group theory...
is an open immersion. Then G is called an open subfunctor of F. If F is covered by representable open subfunctors, then, under certain conditions, it can be shown that F is representable. This is a useful technique for the construction of ringed spaces. It was discovered and exploited heavily by Alexandre Grothendieck, who applied it especially to the case of scheme
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
s. For a formal statement and proof, see Grothendieck, Elements de Geometrie Algebrique, vol. 1, 2nd ed., chapter 0, section 4.5.