What are morphisms of functors

I am not been able to understand, what is a morphism between two functors. I have gone through the formal definition involving a commutative diagram. Can someone explain that to me in a bit more details or in pictures?

Solution 1:

Given categories $\mathcal{C}$ and $\mathcal{D}$, a functor $F:\mathcal{C}\to\mathcal{D}$ consists of two pieces of information:

• A rule assigning, to each object $x\in \mathrm{ob}(\mathcal{C})$, a corresponding $Fx\in\mathrm{ob}(\mathcal{D})$.
• A rule assigning, to each morphism $f\in\mathrm{Hom}_{\mathcal{C}}(x,y)$, a corresponding $Ff\in\mathrm{Hom}_{\mathcal{D}}(Fx,Fy)$.

"Pictorally": $${\mathrm{ob}(\mathcal{C})\xrightarrow{F\text{ (on objects)}}\mathrm{ob}(\mathcal{D})\atop x\;\;\;\;\;\;\;\;\to\;\;\;\;\;\;\;\; Fx}\qquad\qquad {\mathrm{Hom}_{\mathcal{C}}(x,y)\xrightarrow{F\text{ (on morphisms)}}\mathrm{Hom}_{\mathcal{D}}(Fx,Fy)\atop \begin{pmatrix} x\\ \downarrow\!\scriptsize f\\ y\end{pmatrix} \;\;\;\;\;\;\;\;\to\;\;\;\;\;\;\;\;\begin{pmatrix} Fx\\ \downarrow\!\scriptsize Ff\\ Fy\end{pmatrix} }$$ Of course, there are also properties that these rules must have, to ensure that we have some consistency: we must have $F(\mathrm{id}_x)=\mathrm{id}_{Fx}$ and $F(g\circ f)=Fg\circ Ff$. We can't just haphazardly move objects and morphisms around without respecting composition.

Now, what would a "morphism of functors" be? We need a sort of "meta-rule"; a rule for turning the rules defining $F$ into the rules defining $G$.

$$\begin{array}{ccc} & & Fx\\ & \nearrow & \color{red}{\downarrow}\\ x & & \color{red}{\downarrow}\\ & \searrow & \color{red}{\downarrow}\\ & & Gx \end{array}\qquad\qquad \begin{array}{ccc} & & (Fx\xrightarrow{Ff}Fy)\\ & \nearrow & \color{red}{\downarrow}\\[-0.4in] (x\xrightarrow{f} y) & & \color{red}{\downarrow}\\ & \searrow & \color{red}{\downarrow}\\ & & (Gx\xrightarrow{Gf}Gy) \end{array}$$

So, to specify a morphism of functors (let's call it $\alpha$), we can figure that we'll need to do two things:

• for each $x\in\mathrm{ob}(\mathcal{C})$, we want to choose a morphism $\alpha_x:Fx\to Gx$
• for each $f\in\mathrm{Hom}_\mathcal{C}(x,y)$, we want to choose a "map of arrows" from $Ff$ to $Gf$. What is a map of arrows? Why, it's a choice of arrows on each end that make it a commutative square. And hey, we've already chosen maps $\alpha_x:Fx\to Gx$ and $\alpha_y:Fy\to Gy$, so we actually don't need to be choosing anything more; we just add the requirement that the $\alpha_x$'s we chose make a commutative diagram $$\begin{array}{ccc} Fx & \xrightarrow{Ff} & Fy\\ \scriptsize\alpha_x\!\normalsize\downarrow & & \downarrow \!\scriptsize\alpha_y\\ Gx& \xrightarrow[Gf]{} & Gy \end{array}$$ for any morphism $f$ in the category $\mathcal{C}$.

This choice of arrows $\alpha_x$ for each $x\in\mathrm{ob}(\mathcal{C})$ defines a natural transformation $\alpha$ from $F$ to $G$.

There is actually a very deep connection between the notion of natural transformation, and the notion of homotopy (from the field of topology). In the gif animation below (taken from Wikipedia), what we start with are two different plane curves, $f:X\to \mathbb{R}^2$ and $g:X\to\mathbb{R}^2$, where $X$ stands for some closed interval. A homotopy is a "morphing" of one curve to the other.

But what does this really mean? For each $s\in X$, we have chosen a path $h_s:I\to \mathbb{R}^2$ that starts at $f(s)$ and ends at $g(s)$ ($I$ stands for the interval $[0,1]$). Using $t$ to denote the variable for these $h_s$'s, you can think of $t$ as time: at time $t=0$, we have $h_s(0)=f(s)$, and at time $t=1$, we have $h_s(1)=g(s)$, and the curve $h_s(t)$ is the path traced out as $f(s)$ moves to $g(s)$. Of course, there must be some consistency; nearby points on one curve can't have too wildly different paths, the maps $h_s$ must "continuously vary" with $s$.

Sounds familiar! In fact, there is a category analogous $I$, let's call it $\mathcal{I}$: it has two objects, named $0$ and $1$, and it has one non-identity morphism $T:0\to 1$. To be poetic, we might call it the "arrow of time". Just as a homotopy from a map $f:X\to Y$ to a map $G:X\to Y$ is a map $h:X\times I\to Y$ such that $h(x,0)$ is $f(x)$, $h(x,1)$ is $g(x)$, and for each $x\in X$, $h(x,t)$ is traces out a curve from $f(x)$ to $g(x)$, a natural transformation $\alpha$ from a functor $F:\mathcal{C}\to\mathcal{D}$ to a a functor $G:\mathcal{C}\to\mathcal{D}$ can be identified with (or even alternatively defined as) a functor $H:\mathcal{F}\times \mathcal{I}\to\mathcal{G}$, where $H(x,0)=Fx$, and $H(x,1)=Gx$, and $H(x,T)=\alpha_x$.

See this MathOverflow thread Natural transformations as categorical homotopies and this nLab article geometric realization of categories.

Solution 2:

Rather than give you an unproductive definition, I would suggest you to take a look at the paragraph I.4 of Saunders Mac Lane's Categories for the Working Mathematician, where the author writes about natural transformations (morphisms of functors) and give (so to say) an intuitive idea of them as ways of naturally translating a diagram into a category $C$ (given by a functor with codomain $C$) into another. He also spends some time in explaining some examples which might help you in understanding deeper and better the concept (in particular, I'd suggest you to try and solve exercise number 6 at the end of that paragraph: I found it quite useful to comprehend the idea behind natural transformations!). I hope this might help.

Solution 3:

A natural transformation is essentially an "isomorphism" between the images of two functors, our examples being $F,G:{\cal C}\to{\cal D}$. This "isomorphism" is realized by connecting those objects in $\cal D$ in the image of $F$ to those in the image of $G$ via morphisms that already exist in $\cal D$, with the attendant condition that combining any of the morphisms or objects in $\cal D$ at play (meaning, in addition to the objects and morphisms in the image of $F$ and $G$, the "connecting" morphisms as well) one obtains a commutative diagram.

Perhaps a better way of thinking about it, as Marco says, is as "translating" the image of one functor to the image of another functor via morphisms within the codomain.

You can see the image of $\cal C$ under $F$ and $G$ in red and blue respectively over in $\cal D$ (our picture has "zoomed in" onto a particular "region" in the category $\cal D$). The "connecting" morphisms (denoted by $\eta_{\rm obj}$s in the Wikipedia article on natural transformations) are light purple above. The fact that $F$ and $G$ are functors is seen by the fact that all of the objects and arrows in their image are put together in (red and blue) commutative diagrams. The fact that the system of light purple arrows are a natural transformation $F\cong G$ is seen by the fact that they come together with the red and blue data and also still form commutative diagrams.