Functors (Pt. I)
Point of Post: In this post we discuss the notion of functors between categories, and give examples of such functors.
We are now going to discuss probably one of the most fundamental, and influential ideas in the entirety of category theory: functors. In fact, when functors were the reason Mac Lane et. al introduced the notion of categories–they were merely the necessary background to describe functors. So, what are these magical objects, these functors? Intuitively, functors allow us to make rigorous statements such as “by turning a problem in topology into a problem in group theory”. Functors allow us to naturally carry ideas from one category (for all intents and purposes, subject of study) to another. They allow us to make mathematically precise the interconnections and interplay between the various and (artificially) disparate branches of mathematics. Most of the game-changing mathematics in the last fifty years is, in some way or another, traceable back to a functor. Of course, this is definitively hyperbole but it’s impossible to stress this point enough. Functors are, in a crude sense, the “morphisms” in the “category of categories”. For those in the know, it is well-known that this approach has difficulties, but it roughly gives the idea that functors are the structure preserving “maps” between categories. While I could go on, and on, and on about how important functors are, I believe that the best way to make clear how important and pervasive functors are, is to see some examples. Thus, let’s get on with it.
Let and be two categories. Then, a covariant functor from is a map which assigns to each an object and to every morphism in a morphism in subject to the two following conditions a) for every and b) for every in . Thus, we roughly see that functors are “maps” between categories that respect the categories’ structure. Let us now look at some general examples of functors.
Let be a category and a subcategory of . Then, the obvious inclusion is a functor.
The identity functor on a category is the obvious map that is identity on and for every in .
Fix a category and an object . We can then consider the covariant Hom functor which takes and maps it to and maps to where is the map given by . Let us verify that this is, in fact, a functor. To check that for every we merely note that for any we have so that the desired equality follows. We must now check that if then . To do this we note that for any we have that and so the desired equality follows.
Let be a category and . We can then consider the constant functor defined by for every and for every in .
Let’s now take a look at some more, specific examples:
Suppose we have a monoid . We have then discussed how can be viewed as a category with , , and the composition of elements of is just their multiplication. Suppose then that we have another monoid and we consider its corresponding category . Then, a functor is really nothing more than a monoid homomorphism . Indeed, by design the only thing can act on is and so . Moreover, maps in such a way that and . This is precisely the definition of a monoid homomorphism. Of course, it’s easy to see that we can make the similar jump and note that if we consider two groups and as categories with one object, then functors between them are nothing more than group homomorphisms.
There is a natural functor (groups to abelian groups), which is none other than the abelianization functor. To be more specific, let us denote this functor by . Then, for one has that is nothing more than the abelianization of , and given in we can get a map by noting that by composing with the canonical projection we get a map and by the definition of the abelinazation (since is abelian) this factors through to give a map .
As we have discussed previously on this blog there is a functor which takes a ring to its group of units and takes a unital ring homomorphism to the underlying group homomorphism .
There is a functor which sends and to the map sending to .
There is the power-set functor defined by and given we have the induced map where (i.e. the image of under ).
We have seen that if we consider the covariant Hom functor on we actually get a functor and if is commutative we get a functor .
If we fix a ring we have a functor which associates to a set the free module and to every set map the guarnateed extension -map from the set map .
Given a category which is “concrete” (i.e. sets with structure) one can consider forgetful functors where are sets with “less structure” which just “forgets” the structure. For example, there is a forgetful functor which takes a group to its underlying set, and a group homomorphism to its underlying map. Or, there is a forget functor which takes a ring to its underlying abelian group structure, and every ring homomorhpism to the underlying group homomorphism.
If we look at the category of pointed topological space we have the fundamental group functor which associates to each pointed space the fundamental group and to each pointed map the (usual) induced homomorphism .
Consider the functor which sends to , the endomorphism ring of the group.
 Mac, Lane Saunders. Categories for the Working Mathematician. New York: Springer-Verlag, 1994. Print.
 Adámek, Jirí, Horst Herrlich, and George E. Strecker. Abstract and Concrete Categories: the Joy of Cats. New York: John Wiley & Sons, 1990. Print.
 Berrick, A. J., and M. E. Keating. Categories and Modules with K-theory in View. Cambridge, UK: Cambridge UP, 2000. Print.
 Freyd, Peter J. Abelian Categories. New York: Harper & Row, 1964. Print.
 Rotman, Joseph J. Introduction to Homological Algebra. Springer-Verlag. Print.