# What is a Natural Transformation? Definition and Examples

/I hope you have enjoyed our little series on basic category theory. (I know I have!) This week we'll close out by chatting about natural transformations which are, in short, a nice way of moving from one functor to another. If you're new to this mini-series, be sure to check out the very first post, *What is Category Theory Anyway?* as well as *What is a Category?* and last week's *What is a Functor?*

...the notion of category is best excused as that which is necessary in order to have the notion of functor. But the progression does not stop here. There are maps between functors, and they are called natural transformations. And it was in order to define these that Eilenberg and Mac Lane first defined functors.

- Peter J. Freyd

## What is a natural transformation?

**natural transformation**$\eta:F\Longrightarrow G$ from $F$ to $G$ consists of some data that satisfies a certain property.

### The Data

- a morphism $F(x)\overset{\eta_x}{\longrightarrow}G(x)$ for each object $x$ in $\mathsf{C}$

*all*the morphisms $\eta_x$, so sometimes you might see the notation $$\eta=(\eta_x)_{x\in\mathsf{C}},$$ where each $\eta_x$ is referred to as a

*component*of $\eta$. This is very similar to how a sequence $s$ is comprised of the totality of its terms $s=\{s_n\}_{n\in\mathbb{N}}$ or how a vector $\vec{v}$ is comprised of all of its components $\vec{v}=(v_1,v_2,\ldots).$

### The Property

- Whenever $x\overset{f}{\longrightarrow}y$ is a morphism in $\mathsf{C}$, $$G(f)\circ \eta_x=\eta_y\circ F(f).$$ In other words, the square below commutes.

*commute*with the arrows in the diagrams. For example, in the picture below, the black arrows below comprise a natural transformation between two functors* $F$ and $G$.

or, cleaning things up a bit,

To get a better feel for natural transformations, let's look at a few special cases.

## Case #1: F and G are constant

## Case #2: F is constant

For good reasons, $\eta$ in this case is called a **cone over $G$**.

## Case #3: G is constant

Not surprisingly, this type of $\eta$ is called a **cone under $F$** (or sometimes a **cocone**).

**limits**and

**colimits.**You've no doubt come across a (co)limit or two, though perhaps without knowing it. The empty set, the one point set, the intersection, union, and product of sets, the kernel of a group, the quotient of a topological space, the direct sum of vector spaces, the free product of groups, the pullback of a fiber bundle, inverse limits and direct limits are all examples of either a limit or a colimit. Each is special in that it forms a "universal" cone over a particular functor/diagram!

This deserves much more than a few sentences of attention, so we'll chat about more (co)limits in a future post.

## Case 4: each ηₓ is an isomorphism

*up to a change in perspective*. When this is the case, the natural transformation $\eta$ is called a

**natural isomorphism**, and $F$ and $G$ are said to be

**naturally isomorphic.**

## Example #1: group actions & equivariant maps

So what's a natural transformation in this setup? Suppose $A,B:\mathsf{B}G\to\mathsf{Set}$ are two functors with $A(\bullet)=X$ and $B(\bullet)=Y$ and let $\bullet\overset{g}{\longrightarrow}\bullet$ be a group element in $G$. Then $\eta:A\Longrightarrow B$ consists of exactly one function $\eta:X\to Y$ that satisfies $\eta(g(x))=g(\eta(x))$ for every $x\in X$.

This equality follows from the commuting square below. As with all such diagrams, simply pick an element in one of the corners (here, I've picked a little blue $x\in X$, top-left) and chase it around. Notice, the two horizontal maps are exactly the same $X\overset{\eta}{\longrightarrow}Y$, despite the fact that they're drawn in two different locations on the screen.

*same*as first sending $x$ to $Y$ via $\eta$ and then translating that point by $g$. In short, natural transformations are $G$-equivariant maps!

I have a few more more examples to share, but I'll save them until next time. Check back in a few days!

*indexing*category