# Abstract Nonsense

## Functors (Pt. I)

Point of Post: In this post we discuss the notion of functors between categories, and give examples of such functors.

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Motivation

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.

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Functors

Let $\mathcal{C}$ and $\mathcal{D}$ be two categories. Then, a covariant functor $F$ from $\mathcal{C}\to\mathcal{D}$ is a map which assigns to each $C\in\text{obj}(\mathcal{C})$ an object $F(C)\in\text{obj}(\mathcal{D})$ and to every morphism $A\xrightarrow{f} B$ in $\mathcal{C}$ a morphism $F(A)\xrightarrow{F(f)}F(B)$ in $\mathcal{D}$ subject to the two following conditions a) $F(1_A)=1_{F(A)}$ for every $A\in\text{obj}(\mathcal{C})$ and b) $F(g\circ f)=F(g)\circ F(f)$ for every $A\xrightarrow{f}B\xrightarrow{g}C$ in $\mathcal{C}$. 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.

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Let $\mathcal{C}$ be a category and $\mathcal{S}$ a subcategory of $\mathcal{C}$. Then, the obvious inclusion $I:\mathcal{S}\hookrightarrow\mathcal{C}$ is a functor.

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The identity functor $1_\mathcal{C}$ on a category $\mathcal{C}$ is the obvious map that is identity on $\text{obj}(C)$ and $1_C(f)=f$ for every $A\xrightarrow{f}B$ in $\mathcal{C}$.

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Fix a category $\mathcal{C}$ and an object $A\in\text{obj}(\mathcal{C})$. We can then consider the covariant Hom functor $\text{Hom}(A,\bullet):\mathcal{C}\to\mathbf{Set}$ which takes $B\in\text{Obj}(\mathcal{C})$ and maps it to $\text{Hom}(A,B)$ and maps $B\xrightarrow{j}C$ to $\text{Hom}(A,B)\xrightarrow{j_\ast}\text{Hom}(A,C)$ where $j_\ast$ is the map given by $j_\ast(g)=j\circ g$. Let us verify that this is, in fact, a functor. To check that $1_B^\ast=1_{\text{Hom}(A,B)}$ for every $B\in\text{Obj}(\mathcal{C})$ we merely note that for any $g\in\text{Hom}(A,B)$ we have $1_B^\ast(g)=1_B\circ g=g$ so that the desired equality follows. We must now check that if $B\xrightarrow{j}C\xrightarrow{k}D$ then $(k\circ j)_\ast=k_\ast\circ j_\ast$. To do this we note that for any $g\in\text{Hom}(B,D)$ we have that $(k\circ j)_\ast(g)=k\circ j\circ g=k_\ast(j\circ g)=k_\ast(j_\ast(g))$ and so the desired equality follows.

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Let $\mathcal{C}$ be a category and $A\in\text{Obj}(\mathcal{C})$. We can then consider the constant functor $c_A:\mathcal{C}\to\mathcal{C}$ defined by $c_A(B)=A$ for every $B\in\text{obj}(\mathcal{C})$ and $c_A(f)=1_A$ for every $B\xrightarrow{f}C$ in $\mathcal{C}$.

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Let’s now take a look at some more, specific examples:

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Suppose we have a monoid $(M,\ast)$. We have then discussed how $(M,\ast)$ can be viewed as a category $\mathcal{M}$ with $\text{obj}(\mathcal{M})=\{M\}$, $\text{Hom}(M,M)=M$, and the composition of elements of $M$ is just their multiplication. Suppose then that we have another monoid $\left(N,\star\right)$ and we consider its corresponding category $\mathcal{N}$. Then, a functor $F:\mathcal{M}\to\mathcal{N}$ is really nothing more than a monoid homomorphism $M\to N$. Indeed, by design the only thing $F$ can act on is $M$ and so $F(M)=N$. Moreover, $F$ maps $M=\text{Hom}(M,M)\to\text{Hom}(N,N)=N$ in such a way that $F(m\ast m')=F(m)\star F(m')$ and $F(1_M)=1_N$. 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 $(G,\ast)$ and $(H,\star)$ as categories with one object, then functors between them are nothing more than group homomorphisms.

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There is a natural functor $\mathbf{Grp}\to\mathbf{Ab}$ (groups to abelian groups), which is none other than the abelianization functor. To be more specific, let us denote this functor by $^{\text{ab}}$. Then, for $G\in\text{Obj}(\mathbf{Grp})$ one has that $G^{\text{ab}}$ is nothing more than the abelianization of $G$, and given $G\xrightarrow{f}H$ in $\text{Grp}$ we can get a map $G^{\text{ab}}\to H^{\text{ab}}$ by noting that by composing with the canonical projection $\pi:H\to H^{\text{ab}}$ we get a map $G\to H^{\text{ab}}$ and by the definition of the abelinazation (since $H^{\text{ab}}$ is abelian) this factors through $G^{\text{ab}}$ to give a map $G^{\text{ab}}\xrightarrow{f^{\text{ab}}}H^{\text{ab}}$.

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As we have discussed previously on this blog there is a functor $^\times:\mathbf{Ring}\to\mathbf{Grp}$ which takes a ring $R$ to its group of units $R^\times$ and takes a unital ring homomorphism $R\xrightarrow{f}S$ to the underlying group homomorphism $R^\times\xrightarrow{f}S^\times$.

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There is a functor $\mathbf{Ring}\to\mathbf{Ring}$ which sends $R\mapsto R[x]$ and $R\xrightarrow{f}S$ to the map $R[x]\to S[x]$ sending $\displaystyle \sum_k a_kx^k$ to $\displaystyle \sum_k f(a_k)x^k$.

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There is the power-set functor $\mathscr{P}:\mathbf{Set}\to\mathbf{Set}$ defined by $\mathscr{P}(X)=2^X$ and given $X\xrightarrow{f}Y$ we have the induced map $2^X\xrightarrow{\mathscr{P}(f)}2^Y$ where $\mathscr{P}(f)(A)=f(A)$ (i.e. the image of $A$ under $f$).

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We have seen that if we consider the covariant Hom functor on $R\text{-}\mathbf{Mod}$ we actually get a functor $R\text{-}\mathbf{Mod}\to\mathbf{Ab}$ and if $R$ is commutative we get a functor $R\text{-}\mathbf{Mod}\to R\text{-}\textbf{Mod}$.

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If we fix a ring $R$ we have a functor $F:\mathbf{Set}\to R\text{-}\mathbf{Mod}$ which associates to a set $S$ the free module $R[S]$ and to every set map $S\xrightarrow{f}T$ the guarnateed extension $R$-map $R[S]\to R[T]$ from the set map $S\xrightarrow{f}T\hookrightarrow R[T]$.

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Given a category $\mathcal{C}$ which is “concrete” (i.e. sets with structure) one can consider forgetful functors $U:\mathcal{C}\to\mathcal{D}$ where $\mathcal{D}$ are sets with “less structure” which just “forgets” the structure. For example, there is a forgetful functor $U:\mathbf{Grp}\to\mathbf{Set}$ which takes a group to its underlying set, and a group homomorphism to its underlying map. Or, there is a forget functor $U:\mathbf{Ring}\to\mathbf{Ab}$ which takes a ring to its underlying abelian group structure, and every ring homomorhpism to the underlying group homomorphism.

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If we look at the category $\textbf{Top}_\bullet$ of pointed topological space we have the fundamental group functor $\textbf{Top}_\bullet\to\textbf{Grp}$ which associates to each pointed space $(X,x_0)$ the fundamental group $\pi_1(X,x_0)$ and to each pointed map $f:(X,x_0)\to (Y,y_0)$ the (usual) induced homomorphism $f_\ast:\pi_1(X,x_0)\to\pi_1(Y,y_0)$.

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Consider the functor $\textbf{Ab}\to\textbf{Ring}$ which sends $A$ to $\text{End}(A)$, the endomorphism ring of the group.

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A group action of the group $G$ can be considered as nothing more than a functor $F:\mathcal{G}\to\mathbf{Set}$ where $\mathcal{G}$ is the one-element category associated to $G$. Similarly, any complex group representation can be seen as a functor $R:\mathcal{G}\to\mathbf{Vect}_\mathbb{C}$.

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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.

December 27, 2011 -

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