Functor Categories (Pt. II)
Point of Post: This is a continuation of this post.
Properties of Functor Categories
Let’s now look at some of the properties inherited by functor categories from their source and target categories. The first thing we’d like to note is that much of the “work” in functor categories can be thought of being done “componentwise” i.e. by examining individual objects in the source category. As a first homage to this idea we have the following:
Theorem: Let be a small category and a category. Then, if is a morphism of functors in then if enjoys any of the following listed properties as a morphism in for all then enjoys that property as a morphism in :
Proof: Suppose that is monic for each object of and suppose that are natural transformations, where is a functor, such that . Then, by definition, for each object of we have that , now since is monic for each we can conclude that for all and so . Since were arbitrary we may conclude that is monic as desired.
Suppose now that is an isomorphsim for all . For each let be the morphism such that and , we claim that is a natural transformation. In other words, we have to just check that for each morphism in we have that , but this follows by hitting both sides of the known identity by on the left and on the right. Thus, really is a natural transformation, and since and we see that is an isomorphism , and so the conclusion follows.
The other parts of the theorem follow similarly.
Theorem: Let be a small category and a category. Then, if has an initial, terminal, or zero object then so does .
Proof: Let be an initial object of and consider the constant functor . We claim that is an initial object in . Indeed, let be a functor for each object in let be the unique arrow . We claim that is a natural transformation. Indeed, suppose that is an arrow in we then must prove that , or that , but since is an arrow it must be equal to by the definition of initial object, and so naturality follows. Thus, we have at least one morphism , moreover it’s clear it’s the only one since any other morphism would have to give a different arrow for some , contradicting the initial property of . Thus, is initial in as desired.
The case for terminal and zero objects follows similarly by considering the same cosntant functors.
As mentioned in an earlier post, we see that the process of taking functor categories preserves any -category structure the target category might have. In particular, suppose that we are given an -category and a small category . To make into an abelian category, we need to be able to add natural transformations. Namely, given two functors we know that is and so turning this into an abelian group amounts to the addition of natural transformations . This is simple though, because we can just add “object wise”. Namely, suppose we have two natural transformations , we can then add them by declaring where is the predefined addition on . We need to prove that this really defines a natural transformation . Since really is an arrow it suffices to prove naturality. To this end suppose that is an arrow in we need to verify that . But,
so that defined by really is a natural transformation . To check that this notion of the addition of natural transformations defines an -category structure on it suffices to check that but this follows immediately since this is true “object wise” (i.e. ). From the above and our previous observation about the preservation of zero objects we have the following:
Theorem: Let be a small caetgory and an -category. Then, with addition of natural transformations is an -category. Moreover, if is preadditive then so is .
If we further suppose that is an -category it’s easy to see that:
Theorem: Let be a small -category, and an -category. Then, the category of all additive functors is an -category. Moreover, if is preadditive then so is .
Proof: We know that is a full subcategory of from where it follows that is an -category (since full subcategories of -categories are -subcategories). To prove that is preadditive, assuming , is equivalent to checking that the constant functor for an initial object of (as described above) is additive. But, this follows immediately since if are arrows then since , as previously discussed, is the additive identity of .
We discuss two last things which come up often when discussing functor categories. The first is the evaluation functor associated to a functor category. Namely, suppose we are given some functor category , there is then an obvious way to construct a functor defined by and given morphisms in and in we define . It’s easy to verify that this is actually a functor.
The second bit of miscellany I’d like to mention is the following theorem:
Theorem: Let be two small categories and a category. Then, .
The proof of this is annoying, but fairly straightforward. It can be found on page 97 of .
 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.
 Herrlich, Horst, and George E. Strecker. Category Theory: An Introduction. Lemgo: Heldermann, 2007. Print.
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