Product of Categories (Pt. II)
Point of Post: This is a continuation of this post.
What we’d now like to discuss is what we use to motivate the creation of the product of categories. Namely, although the last theorem has to do with functors into product categories we’d now like to discuss functors out of product categories. Namely, given categories we call a functor a bifunctor. The main thing one wants to discuss about bifunctors is in relation to the functors one gets when one holds fixed on of the coordinates of the functor. For example, it seems likely that given any category there is going to be a bifunctor which is intimately related to the two functors and . In this case, it’s fairly easy to see that is uniquely determined by the family of functors and as vary over the objects of . So the question is, in general, given a bifunctor is uniquely determined by the functors and , and moreover whether given a paramaterized set of functors and (for every objects in respectively) whether there exists a functor such that and for each –moreover, is unique with respect to this property? The following theorem completely answers this question, but we may first, without any thought see there has to be a restriction on these functors. Namely, if these paramaterized family of functors are supposed to be the result of taking a bifunctor and fixing entries, then we must obviously have for all objects so from the outset we see that these paramaterized family of functors have to satisfy this symmetry condition.
Theorem (Fundamental Theorem of Bifunctors): Let be categories and suppose we have a paramaterized set of functors and , where are objects of respectively, such that for all . Then there exists a bifunctor such that and if and only if for every pair of morphisms in and in one has the following commutative diagram:
Moreover, if such a bifunctor exists, it is unique.
Proof: Let’s talk about necessity of this diagram. While at first this may “look scary” how I always remember this diagram, how I actually was able to write the above one out without reference, is just pretending that the and are the fixed-entry functors resulting from some bifunctor . I then wrote down the two different ways of getting from to and noted that this diagram must be commutative, and then at each point used the fact that . The diagram I’m talking about is, of course,
In fact, it should be clear from that heuristic precisely how to complete the argument–the same way! In other words, the heuristic is the full argument, because we are assuming there exists such a bifunctor!
Conversely, suppose that we have a paramaterized set of functors looking like and for each object in and each object in such that they satisfy the symmetry condition, and satisfy the diagram . We then define to act on objects by and to act on morphisms by taking to which is equal to . The fact that this is a functor but simple laborious diagram chasing.
Of course, as is also true in most algebraic settings, if we have categories and functors and then we can form a bifunctor by sending and . This is called the product of the functors and .
Natural Transformations Between Bifunctors
An obvious question, especially considering the material discussed in the fundamental theorem of bifunctors, is whether an assignment of mappings for bifunctors such that and are natural transformations and is necessarily a bifunctor . This turns out to be the case, as the following shows:
Theorem: The assigment of morphisms is a natural transformation for functors if and only if the assignment of mappings and define natural transformations and for all objects in and objects in . Moreover, is a natural equivalence if and only if each and is.
Proof: Suppose first that and are functors for each set of objects in their respective categories and consider the assigment . What we need to prove is that given a morphism in that . But:
The rest of the theorem follows similarly, just in reverse.
We list here a few facts which, while important, are trivial to prove and are left to anyone interested in the proof:
 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.