What is a Functor? Definition and Examples, Part 1

Next up in our mini series on basic category theory: functors! We began this series by asking What is category theory, anyway? and last week walked through the precise definition of a category along with some examples. As we saw in example #3 in that post, a functor can be viewed an arrow/morphism between two categories.

...every sufficiently good analogy is yearning to become a functor.

- John Baez

What is a functor?

More precisely, a functor $F:\mathsf{C}\to\mathsf{D}$ from a category $\mathsf{C}$ to a category $\mathsf{D}$ consists of some data that satisfies certain properties.

The Data

• an object $F(x)$ in $\mathsf{D}$ for every object $x$ in $\mathsf{C}$
• a morphism $F(x)\overset{F(f)}{\longrightarrow}F(y)$ in $\mathsf{D}$ for every morphism $x\overset{f}{\longrightarrow}y$ in $\mathsf{C}$

The Properties

• $F$ respects composition, i.e. $F(g\circ f)=F(g)\circ F(f)$ in $\mathsf{D}$ whenever $g$ and $f$ are composable morphisms in $\mathsf{C}.$
• $F$ sends identities to identities, i.e. $F(\text{id}_x)=\text{id}_{F(x)}$ for all objects $x$ in $\mathsf{C}.$

Example #1: a functor between groups

We noted last week that every group $G$ can be viewed as a one-object category called $\mathsf{B}G$. Suppose then that $\mathsf{B}G$ and $\mathsf{B}H$ are two such categories. What would a functor $F:\mathsf{B}G\to\mathsf{B}H$ look like?           

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To start, it must send the single object ${\color{RubineRed}\bullet}$ of $\mathsf{B}G$ to the single object ${\color{ProcessBlue}\bullet}$ of $\mathsf{B}H$. Moreover, any morphism ${\color{RubineRed}\bullet}\overset{g}{\longrightarrow} {\color{RubineRed}\bullet}$, which, you'll remember, is just a group element $g\in G$, must map to a morphism ${\color{ProcessBlue}\bullet}\overset{F(g)}{\longrightarrow} {\color{ProcessBlue}\bullet}$. That is, $F(g)$ must be an element of $H$.  The composition property requires that $F(g\circ g')=F(g)\circ F(g')$ for all group elements $g,g'\in G$. And finally, if ${\color{RubineRed}e}$ denotes the identity morphism on ${\color{RubineRed}\bullet}$ (i.e. the identity element in $G$), then we must have $F({\color{RubineRed}e})={\color{ProcessBlue}e}$ where ${\color{ProcessBlue}e}$ is the identity morphism on ${\color{ProcessBlue}\bullet}$ (i.e. the identity element in $H$).    

So what's a functor $F:\mathsf{B}G\to\mathsf{B}H$? It's precisely a group homomorphism from $G$ to $H$!          

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So in this example, a functor is just a  function (which happens to be compatible with the group structure). But what if the domain/codomain of a functor has more than one object in it?  

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Example #2: the fundamental group

There is a functor $\pi_1:\mathsf{Top}\to\mathsf{Group}$ that associates to every topological space* $X$ a group $\pi_1(X)$, called the fundamental group of $X$, and which sends every continuous function $X\overset{f}{\longrightarrow}Y$ to a group homomorphism $\pi_1(X)\overset{\pi_1(f)}{\longrightarrow}\pi_1(Y)$.  

The elements of $\pi_1(X)$ are (homotopy classes of) maps of the circle into the space $X$. But don't worry if you're not familiar with the phrase "homotopy classes of." Just know that informally $\pi_1$ is a "hole-detector"---it keeps track of the various loops in $X$. And knowing that $\pi_1$ is functorial allows you to prove cool things like Brouwer's fixed point theorem. For instance, see the proof of Theorem 1.3.3 here.   Now it turns out that the fundamental group is an invariant of a topological space. In other words, if $X$ and $Y$ are homeomorphic spaces then their fundamental groups are isomorphic. (Even better, this sentence is still true if we replace "homeomorphic" by "homotopic" which is a slightly weaker notion of "sameness.") This gives us a handy way to distinguish spaces: if $X$ and $Y$ have nonisomorphic fundamental groups, then $X$ and $Y$ are guaranteed to be topologically different.  

(Actually, there's nothing special about the number 1 here. For each $n\geq 1$, there is a functor $\pi_n:\mathsf{Top}\to\mathsf{Group}$ that sends $X$ to $\pi_n(X)$ which is again, informally, the group whose elements are maps of the $n$-dimensional sphere into $X$. The groups $\pi_n(X)$ for $n>1$ are called the higher homotopy groups of $X$. And when $X$ is a sphere, cool and spooky things happen.)

 In practice it can be helpful to think of a functor $F:\mathsf{C}\to\mathsf{D}$ as encoding an invariant of some sort.          

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This is because if two objects $x$ and $y$ are "the same" in $\mathsf{C}$, then $F(x)$ and $F(y)$ must be "the same" in $\mathsf{D}$. By "the same," I'm referring to the precise notion of isomorphic:  

 In any category, a morphism $x\overset{f}{\longrightarrow}y$ is said to be an isomorphism if there exists a morphism $y\overset{g}{\longrightarrow}x$ so that $g\circ f=\text{id}_x$ and $f\circ g=\text{id}_y$.

Isomorphisms in $\mathsf{Set}$ are called bijections, isomorphisms in $\mathsf{Top}$ are called homeomorphisms, isomorphisms in $\mathsf{Man}$ (the category of smooth manifolds and smooth maps) are called diffeomorphisms, isomorphisms in $\mathsf{Vect}_{\mathbb{k}}$ (the category of $\mathbb{k}$-vector spaces with linear transformations) are called, well, isomorphisms, and so on.** So we can rephrase the above more formally: "If $f$ is an isomorphism between $x$ and $y$ then $F(f)$ is an isomorphism between $F(x)$ and $F(y)$." (The proof isn't tricky - it follows directly from the definitions!) Or, to put it simply, functors preserve isomorphisms. Well, I have a couple more examples of functors that I'm excited to share with you---one of them makes an appearance in a familiar equation from calculus!---but I'll save them for tomorrow.

 Stay tuned!

*Technically, they are  pointed topological spaces, i.e.  spaces where you declare a certain point, called a basepoint, to be special/distinguished. So really, $\pi_1$ is from $\mathsf{Top}_*$ to $\mathsf{Group}$ where $\mathsf{Top}_*$ denotes the category of pointed spaces with basepoint-preserving maps as morphisms.

**Fun fact: Categories in which every morphism is an isomorphism are given a special name: groupoids. (Can you guess why?) Every group, then, is precisely a one-object groupoid. And yes! The fundamental groupoid of a topological space is a thing! Perhaps we'll chat about it in a future post....