# A Diagram is a Functor

Last week was the start of a mini-series on limits and colimits in category theory. We began by answering a few basic questions, including, "What *ARE* (co)limits?" In short, they are a way to construct new mathematical objects from old ones. For more on this non-technical answer, be sure to check out Limits and Colimits, Part 1. Towards the end of that post, I mentioned that (co)limits aren't really related to *limits of sequences* in topology and analysis (but see here). There is, however, one similarity. In analysis, we ask for the limit *of* a sequence. In category theory, we also ask for the (co)limit OF* *something. But if that "something" is not a sequence, then what is it?

Answer: a diagram.

We've talked about diagrams before: for a quick refresher, check out this post. Today I'd like to give you a different way to think about diagrams - namely, as functors! In other words, I hope to convince you that

**a diagram is ***a functor*.

*a functor*.

Once we adopt this viewpoint, we'll be ready to look at the formal definition of limits and colimits. Now, how can we view diagrams as functors? Suppose $F:\mathsf{I}\to\mathsf{C}$ is a functor between categories $\mathsf{I}$ and $\mathsf{C}$. We'll call $\mathsf{I}$ an **indexing category**, and for the sake of illustration let's suppose it's a simple one:

I've labeled the objects in $\mathsf{I}$ with colors and there is an identity arrow for each object, though I haven't drawn them. Let's also suppose that the horizontal arrow arrow is the composition of the two diagonal arrows.

So what's a functor $F$ out of this category?

It's simply a choice of three objects and three arrows in $\mathsf{C}$.

Here $F({\color{Magenta}\bullet})=A$ and $F({\color{RoyalBlue}\bullet})=B$ and $F({\color{Green}\bullet})=C$, and the image of the three arrows in $\mathsf{I}$ are the arrows $f,g$ and $h$ in $\mathsf{C}$ where $f=h\circ g$. So you see? That's all there is to it! The image of $F$ is no more and no less than a diagram in $\mathsf{C}$. We might even call it an "$\mathsf{I}$-shaped diagram" since different shapes for $\mathsf{I}$ lend to different shapes of diagrams. For example,

In short, a diagram *is* a functor.

## By the way...

This idea of identifying a *map* with its *image* is nothing new. After all, a *sequence* of real numbers is technically a function $x:\mathbb{N}\to\mathbb{R}$, though we usually write $x_n$ for the image $x(n)$ and think of the sequence as the collection $\{x_n\}_{n\in\mathbb{N}}$ rather than the function $x$ itself.

Likewise the formal definition of a *path* in a topological space $X$ is: "a continuous function from the closed unit interval into $X$," i.e. $p:[0,1]\to X$. But when we think about paths, we often have the image $p(I)\subset X$ of $p$ in mind.

And in differential geometry, a *vector field* on a differentiable manifold $M$ is a section of the tangent bundle, i.e. a map $\phi$ from $M$ into its tangent bundle $TM$ such that the composition of $\phi$ with the projection $TM\to M$ is the identity on $M$. Of course that was a mouthful, and so we often just think of a vector field as a collection of tangent vectors - one attached to each point on the manifold. That is, we identify $\phi$ with its image.
These examples are all similar to the statement, "a diagram is a functor."

## Back to (co)limits...

Now that we can view *diagrams* as *functors,* we can make sense of *maps between diagramas*, i.e. natural transformations between functors. As we'll see next time, the *(co)limit of a diagram $F$* is a particular natural transformation between $F$ and another diagram of a *particular* shape. What's neat is that if $F$ is shaped like one of those diagrams drawn in the table above, then the (co)limit is given a familiar name, like *intersection, union, Cartesian product, kernel, direct sum, and quotient!*

We'll explore all the details in the coming weeks.