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 Set of all sets. 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 1_"S" 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 1_"T" on "T". Notice that 1_"T" is the restriction of 1_"S" 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, and it is usually used when defining Grothendieck topologies.

Open subfunctors

Subfunctors are also used in the construction of representable functors on the category of ringed spaces. 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"×_"F"Hom(−, "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 schemes. For a formal statement and proof, see Grothendieck, "Elements de Geometrie Algebrique", vol. 1, 2nd ed., chapter 0, section 4.5.