# The Category Of Graphs

29 Jun 2019

You know the sensation where a familiar word becomes suddenly completely strange for no apparent reason? Like suddenly the scales fall away from your eyes and you see how utterly weird it is that a concept has a particular name? I’ve been having that feeling with the word graph as I use it in math.

As a kid, you learn how to graph simple functions in the sense of being able to sketch them on a sheet of, say, graph paper. Millions of adults probably faintly remember the shape of $y = x^2$ but aren’t quite sure why.

To mathematicians and computer scientists, though, for some reason graph means something similar, but different: to them a graph is an object with vertices and edges. Indeed, to Serre, that’s literally all a graph is: a quadruple $(V,E,\bar{},\iota)$. Two sets, $V$ and $E$, represent the vertices and (oriented) edges of the graph, and $\bar{}\colon E \to E$ and $\iota\colon E \to V$ are functions; $\bar{e}$ is the edge $e$ with its orientation reversed, (so $\bar{\bar{e}} = e$ for any edge $e$, but $\bar e \neq e$), and $\iota(e)$ represents the initial vertex of the edge $e$. This lets us define a function $\tau\colon E \to V$ picking out the terminal vertex of an edge by setting $\tau(e) = \iota(\bar{e})$.

A topologist would prefer to say that a graph is a 1-dimensional CW complex. So rather than an abstract set of vertices, we have a set of points, $V$, the $0$-cells of our CW complex. And rather than an abstract set of oriented edges, we have a collection of $1$-cells (intervals of the real line) whose endpoints are attached to $0$-cells.

These two descriptions really are in agreement for most purposes. Maybe one would prefer the more combinatorial description as a way of “rigidifying” some of the inherent floppiness of topological spaces. At the same time, using topological methods allows one access to a whole host of techniques that a too-rigid approach might bury under a sea of formalism.

But of course, neither definition really squares with the idea of the graph of $y = x^2$, hence why I’ve been feeling weird about the word.

Anyway, the category of graphs, introduced by John Stallings in Topology of Finite Graphs, building on work of Serre in Trees, more about which almost certainly soon, has (as all categories do) collections of objects and arrows between them. Here the objects are graphs and arrows are morphisms of graphs. A morphism $f\colon G \to H$ between graphs $G$ and $H$ is a pair of functions $f_V\colon V(G) \to V(H)$ and $f_E\colon E(G) \to E(H)$, such that the following diagrams commute: $\require{AMScd}$

In English what this is saying is that $f$ preserves the graph structure: it sends edges to edges, vertices to vertices, $e$ and $\bar e$ are sent to $f(e)$ and $\overline{f(e)}$, and the initial vertex of $e$ is sent to the initial vertex of $f(e)$.

Stallings emphasizes the rigidity of morphisms in this construction:

By graph and map of graphs, I mean something purely combinatorial or algebraic. Pictures can be drawn, but one has to understand that maps are rigid and not just continuous, maps do not collapse edges or wrap edges around several edges.

And it’s true; it’s a little challenging to describe precisely which topological maps between graphs are even homotopic to morphisms without giving the game (Stallings’ paper) away too quickly!