Functor Categories (Pt. I)
Point of Post: In this post we discuss the notion of functor categories. We give the definition of functor categories, prove some elementary theorems concerning them, and show how pervasive they are by showing that many common categories/subjects are nothing more than disguised functor categories.
So far we have discussed a few ways to construct new categories from old ones. For example, given two categories and we can form their product category . In this post we discuss a new way to form a category from and . Namely, we shall see that there is a useful way in which to make the set of all functors into a category, with the morphisms being nothing more than natural transformations. This may not sound at all important at first, but in fact functor categories play a pivotal role in category theoretic subjects. Why? Suppose that we’ve committed ourselves to studying some particular category, maybe , or for some ring . We then attempt to apply one of the codifying techniques of mathematics: to study an object we should study the interaction of with other objects –i.e. we should study . In the case of category theory this roughly translates to studying the functors between two given categories. The morphisms then can be thought of as natural transformations between these functors. As we shall see with the examples below, a very large number of categories can be cast as functor categories.
Let be a small category and a category. We define the functor category from to , denoted either or , to be the category where the objects are functors , the morphisms are natural transformations , and the composition is the usual composition of natural transformations. We claim that this actually is a category. To see this we must essentially show that the composition of two natural transformations is a natural transformation, there is an “identity” natural transformation, the composition of natural transformations is associative. All of these have already been proven, with the identity transformation defined, as per usual, by .
The reason we need to consider the category to be small is purely a set-theoretic nuisance. Namely, we required in the definition of a category that the class of objects really is a class, and if is not small then the functors need not form a class.
If and are both -categories then we define to be the subcategory of consisting of all additive functors .
Let us now give some examples to show how functor categories naturally come up in the day-to-day workings of mathematics.
Let be a group thought of as a category with one object, in the usual way. We can then consider the functor category . Although this may sound a little complicated there really is a very unscary interpretation of this category. Namely, is nothing more than the category of -sets. Namely, we have noted before that a functor is nothing more than a -action. Let’s take a moment to further explain what this means. Since only contains one object, namely itself, we know that the object function associated to the functor takes to some set . Now, the arrows of are nothing more than the elements of so that the arrow function associated to is nothing more than an assignment of elements of to set functions . But, recall that since all the arrows, being elements of , are isomorphisms and since functors preserve isomorphisms we see that, in fact, the arrow function associated to sends elements of is isomorphisms or bijections . From this, we see that the arrow function is actually a mapping where is the (set!) of bijections . What we claim is that if we think of as the usual symmetric group on then the arrow function is actually a group homomorphism. But, this follows immediately from the fact that, by definition of functors, for any . Thus, we can think of a functor as being nothing more than an ordered pair where is some set and is a group homomorphism. But, ordered pairs as just described are precisely -sets, in other words, sets with a distinguished -action defined by .
Ok, so we see now that we can identify elements of with -sets, but what about the arrows of ? Can we interpret these as nothing more than -set homomorphisms? Indeed, suppose we had two elements of , in other words to functors . From the above we can identify and with two -sets and . Suppose now that we had an arrow in from to , in other words a natural transformation . Let’s see how our identifications and interact with . Namely, since only has one object we see that really only involves one function , or . Now, since is a natural transformation we know that given any arrow in , i.e. some group element , we have the following relation , or recasting in terms of our identifications this says that . But, taking any and recalling how our -action is defined, this says that
and so is a -set homomorphism . It should be pretty obvious from this, that if we had defined to be the set of all -sets with -set homomorphisms then via the functor (which is seen to be an isomorphism) taking to .
It’s easy to see, following the above examples, that given any monoid the functor category can be thought of as the category of -sets with -set homomorphisms.
What we’d now like to show is that fixing a unital ring we have that the category of unital modules over is isomorphic to where where is the -category with the single object . Indeed, we have already remarked that an additive functor is nothing more than a unital module with the identification where is the induced mapping . It’s easy to see that this really is an isomorphism of categories.
What we’d now like to take a look at is, given a group thought of as a category with one object where all the arrows are invertible, the functor category where is some field and interpret in a more hospitable way. Namely, similar to the case we see that we can interpret a functor as being equivalent to an ordered pair where is some finite dimensional -space and is a homomorphism . In other words, a functor is nothing more than a finite-dimensional representation of . In particular, we can easily interpret as being the category of finite dimensional -representations of .
If is the set thought of as a discrete category then let us try to figure out what the functor category looks like. We see that the image of a functor really looks like a pair of morphisms and if are two functors we see that a natural transformation is really a pair of arrows and , and that’s it. Indeed, since the only arrows in are the identity arrows which trivially satisfy the naturality condition we see that any pair of arrows gives rise to a natural transformation. Thus, in this way it’s easy to identify with the product category . This also then gives a natural definition for the arbitrary product of a category with itself. Namely, if is any set, then we define the -fold product of to be the functor category where is now thought of as a discrete category.
As a last example we let be the category (i.e. the category with two objects, and two non-trivial arrows going in the same direction), we want to get a better handle of what looks like. We see that a functor is a pair and a pair of functions . We see then that a natural transformation is just a pair of set functions and such that intertwine with by , in other words we have that and . If we think about a functor as being a directed graph (with the functions as being the arrow functions, assinging edges to their corresponding verticles) then we see that a natural transformation is nothing more than a function between vertex sets which agrees with the edge assignments–thus a graph homomorphism. Thus, the category is nothing more than the category of directed graphs.
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