# Abstract Nonsense

## 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.

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Motivation

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So far we have discussed a few ways to construct new categories from old ones. For example, given two categories $\mathcal{C}$ and $\mathcal{D}$ we can form their product category $\mathcal{C}\times\mathcal{D}$. In this post we discuss a new way to form a category from $\mathcal{C}$ and $\mathcal{D}$. Namely, we shall see that there is a useful way in which to make the set of all functors $F:\mathcal{C}\to\mathcal{D}$ 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 $\mathbf{Grp},\mathbf{Set}$, or $R\text{-}\mathbf{Mod}$ for some ring $R$. We then attempt to apply one of the codifying techniques of mathematics: to study an object $X$ we should study the interaction of $X$ with other objects $Y$–i.e. we should study $\text{Hom}(X,Y)$. 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.

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Functor Categories

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Let $\mathcal{S}$ be a small category and $\mathcal{C}$ a category. We define the functor category from $\mathcal{S}$ to $\mathcal{C}$, denoted either $\mathcal{C}^\mathcal{S}$ or $\text{Fun}(\mathcal{S},\mathcal{C})$, to be the category where the objects are functors $F:\mathcal{S}\to\mathcal{C}$, the morphisms are natural transformations $\eta:F\implies G$, 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 $1_F:F\implies F$ defined, as per usual, by $1_F(x)=1_{F(x)}$.

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The reason we need to consider the category $\mathcal{S}$ 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 $\mathcal{S}$ is not small then the functors $F:\mathcal{S}\to\mathcal{C}$ need not form a class.

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If $\mathcal{S}$ and $\mathcal{C}$ are both $\mathbf{Ab}$-categories then we define $\text{Add}(\mathcal{S},\mathcal{C})$ to be the subcategory of $\mathcal{C}^\mathcal{S}$ consisting of all additive functors $\mathcal{C}\to\mathcal{D}$.

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Let us now give some examples to show how functor categories naturally come up in the day-to-day workings of mathematics.

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Let $G$ be a group thought of as a category with one object, in the usual way. We can then consider the functor category $\mathbf{Set}^G$. Although this may sound a little complicated there really is a very unscary interpretation of this category. Namely, $\mathbf{Set}^G$ is nothing more than the category of $G$-sets. Namely, we have noted before that a functor $F:G\to\mathbf{Set}$ is nothing more than a $G$-action. Let’s take a moment to further explain what this means. Since $G$ only contains one object, namely $G$ itself, we know that the object function associated to the functor $F$ takes $G$ to some set $X=F(G)$. Now, the arrows of $G$ are nothing more than the elements of $G$ so that the arrow function associated to $F$ is nothing more than an assignment of elements of $G$ to set functions $X\to X$. But, recall that since all the arrows, being elements of $G$, are isomorphisms and since functors preserve isomorphisms we see that, in fact, the arrow function associated to $F$ sends elements of $G$ is isomorphisms $X\to X$ or bijections $X\to X$. From this, we see that the arrow function is actually a mapping $G\to S_X$ where $S_X$ is the (set!) of bijections $X\to X$. What we claim is that if we think of $S_X$ as the usual symmetric group on $X$ then the arrow function is actually a group homomorphism. But, this follows immediately from the fact that, by definition of functors, $F(gh)=F(g)\circ F(h)$ for any $g,h\in G$. Thus, we can think of a functor $F:G\to\mathbf{Set}$ as being nothing more than an ordered pair $(X,f)$ where $X$ is some set and $f:G\to S_X$ is a group homomorphism. But, ordered pairs $(X,f)$ as just described are precisely $G$-sets, in other words, sets with a distinguished $G$-action defined by $g\cdot x=f(g)(x)$.

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Ok, so we see now that we can identify elements of $\mathbf{Set}^G$ with $G$-sets, but what about the arrows of $\mathbf{Set}^G$? Can we interpret these as nothing more than $G$-set homomorphisms? Indeed, suppose we had two elements of $\mathbf{Set}^G$, in other words to functors $F_1,F_2:G\to\mathbf{Set}$. From the above we can identify $F_1$ and $F_2$ with two $G$-sets $(X_1,f_1)$ and $(X_2,f_2)$. Suppose now that we had an arrow in $\mathbf{Set}^G$ from $F_1$ to $F_2$, in other words a natural transformation $\eta:F_1\implies F_2$. Let’s see how our identifications $F_1\leftrightarrow (X_1,f_1)$ and $F_2\leftrightarrow (X_2,f_2)$ interact with $\eta$. Namely, since $G$ only has one object we see that $\eta$ really only involves one function $\eta_G:F_1(G)\to F_2(G)$, or $\eta_G:X_1\to X_2$. Now, since $\eta$ is a natural transformation we know that given any arrow in $G$, i.e. some group element $g\in G$, we have the following relation $F_2(g)\circ\eta_G=\eta_G\circ F_1(g)$, or recasting in terms of our identifications this says that $f_2(g)\circ\eta_G=\eta_G\circ f_1(g)$. But, taking any $x\in X_1$ and recalling how our $G$-action is defined, this says that

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$\eta_G(g\cdot x)=\eta_G(f_1(g)(x))=f_2(g)(\eta_G(x))=g\cdot \eta_G(x)$

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and so $\eta_G$ is a $G$-set homomorphism $(X_1,f)\to (X_2,f)$. It should be pretty obvious from this, that if we had defined $G\text{-}\mathbf{Set}$ to be the set of all $G$-sets with $G$-set homomorphisms then $G\text{-}\mathbf{Set}\cong \mathbf{Set}^G$ via the functor (which is seen to be an isomorphism) taking $F$ to $(X,f)$.

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It’s easy to see, following the above examples, that given any monoid $M$ the functor category $\mathbf{Set}^M$ can be thought of as the category of $M$-sets with $M$-set homomorphisms.

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What we’d now like to show is that fixing a unital ring $R$ we have that the category $R\text{-}\mathbf{Mod}$ of unital modules over $R$ is isomorphic to $\text{Add}(\mathcal{C}_R,\mathbf{Ab})$ where $\mathcal{C}_R$  where $\mathcal{C}_R$ is the $\mathbf{Ab}$-category with the single object $R$. Indeed, we have already remarked that an additive functor $F:\mathcal{C}_R\to\mathbf{Ab}$ is nothing more than a unital module with the identification $F\leftrightarrow (F(R),f)$ where $f$ is the induced mapping $R\to\text{End}_\mathbb{Z}(F(R))$. It’s easy to see that this really is an isomorphism of categories.

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What we’d now like to take a look at is, given a group $G$ thought of as a category with one object where all the arrows are invertible, the functor category $\mathbf{FinVect}_k^G$ where $k$ is some field and interpret in a more hospitable way. Namely, similar to the case $\mathbf{Set}^G$ we see that we can interpret a functor $F:G\to\mathbf{FinVect}_k$ as being equivalent to an ordered pair $(V,f)$ where $V$ is some finite dimensional $k$-space and $f$ is a homomorphism $G\to\text{GL}(V)$. In other words, a functor $F:G\to\mathbf{FinVect}_k$ is nothing more than a finite-dimensional representation of $G$. In particular, we can easily interpret $\mathbf{FinVect}_\mathbb{C}^G$ as being the category of finite dimensional $\mathbb{C}$-representations of $G$.

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If $X$ is the set $\{0,1\}$ thought of as a discrete category then let us try to figure out what the functor category $\mathcal{C}^X$ looks like.  We see that the image of a functor $F:X\to\mathcal{C}$ really looks like a pair of morphisms $(F(0),F(1))$ and if $F,G$ are two functors $X\to\mathcal{C}$ we see that a natural transformation $\eta:F\implies G$ is really a pair of arrows $\eta_0:F(0)\to G(0)$ and $\eta(1):F(1)\to G(1)$, and that’s it. Indeed, since the only arrows in $X$ 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 $\mathcal{C}^X$ with the product category $\mathcal{C}\times\mathcal{C}$. This also then gives a natural definition for the arbitrary product of a category $\mathcal{C}$ with itself. Namely, if $X$ is any set, then we define the  $X$-fold product of $\mathcal{C}$ to be the functor category $\mathcal{C}^X$ where $X$ is now thought of as a discrete category.

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As a last example we let $\mathcal{S}$ be the category $\bullet_0\overset{\displaystyle \overset{\text{ }}{\longrightarrow}}{\longrightarrow}\bullet_1$ (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 $\mathbf{Set}^\mathcal{S}$ looks like. We see that a functor $F:\mathcal{S}\to\mathbf{Set}$ is a pair $(E=F(\bullet_0),V=F(\bullet_1))$ and a pair of functions $f_1,f_2:E\to V$. We see then that a natural transformation $\eta:F\implies G$ is just a pair of set functions $\eta_1:E_F\to E_G$ and $\eta_2:V_F\to V_G$ such that $f_1,f_2$ intertwine with $g_1,g_2$ by $\eta_2$, in other words we have that $\eta_2\circ f_1=g_1\circ \eta_2$ and $\eta_2\circ f_2=g_2\circ\eta_2$. If we think about a functor $F:\mathcal{S}\to\mathbf{Set}$ as being a directed graph (with the functions $f_1,f_2:E\to V$ 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 $\mathbf{Set}^\mathcal{S}$ is nothing more than the category of directed graphs.

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References:

[1] Mac, Lane Saunders. Categories for the Working Mathematician. New York: Springer-Verlag, 1994. Print.

[2] Adámek, Jirí, Horst Herrlich, and George E. Strecker. Abstract and Concrete Categories: the Joy of Cats. New York: John Wiley & Sons, 1990. Print.

[3] Berrick, A. J., and M. E. Keating. Categories and Modules with K-theory in View. Cambridge, UK: Cambridge UP, 2000. Print.

[4] Freyd, Peter J. Abelian Categories. New York: Harper & Row, 1964. Print.

[5] Rotman, Joseph J. Introduction to Homological Algebra. Springer-Verlag. Print.

[6] Herrlich, Horst, and George E. Strecker. Category Theory: An Introduction. Lemgo: Heldermann, 2007. Print.

January 7, 2012 -

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