Lecture 36 - Categories from Graphs

# Lecture 36 - Categories from Graphs

In Section 3.2.1 of Seven Sketches, Brendan Fong and David Spivak describe a nice way to get categories from graphs. It's very simple, yet important for building databases.

Here's the basic idea. Start with a graph, for example this:

It has a bunch of "edges", like $$f,g,h,i$$, going between "nodes" like $$A,B,C,D$$. Then build a category where the objects are the nodes and the morphisms are the paths of edges. We compose morphisms by attaching one path onto the end of another.

For example $$f$$ is a path of length 1 from $$A$$ to $$B$$, so it gives a morphism $$f: A \to B$$. $$h$$ is a path of length 1 from $$B$$ to $$D$$, so it gives a morphism $$g: B \to D$$. We can compose these two morphisms and get $$h \circ f : A \to D$$, which is a path of length 2 from $$A$$ to $$D$$.

There's also another morphism from $$A$$ to $$D$$, namely $$i \circ g : A \to D$$.

In this example the longest paths have length 2, but in general a path could have any natural number as its length, including 0. Paths of length 0 are important because they give the identity morphisms in our category. For example, there's a path of length 0 from $$A$$ to $$A$$, which you can't see because it has no edges! It gives the identity morphisms $$1_A: A \to A$$.

When we build a category from a graph this way, it's called the free category on that graph. And if we call the graph $$G$$, we call the free category on that graph $$\mathbf{Free}(G)$$.

I should warn you that different people mean different things by "graph", depending on:

• whether we put arrows on the edges,
• whether we allow more than one edge going from one node to another,
• whether we allow edges going from a node to itself,
• whether the edges have names, and
• whether we allow infinitely many nodes and edges.

But category theorists have a very specific thing in mind when we say "graph". We are very generous: we say yes to all these questions!

So, for example, we allow this graph:

This graph has just one edge. But the free category on this graph has infinitely many morphisms, namely [ 1_z : z \to z ] [ s : z \to z ] [ s \circ s : z \to z ] [ s \circ s \circ s : z \to z ] and so on. There's one morphism for each natural number. In fact, this is how category theorists often think about the natural numbers!

We also allow this graph:

Puzzle 104. How many paths of length $$n$$ go from $$x$$ to $$x$$ in this graph?

The answer is a famous sequence of numbers. So, you're getting a famous sequence from the free category on a graph!

It may sound complicated to let our graphs have so many features, but it's actually more complicated to disallow them. The category theorists' definition of graph is very simple:

Definition. A graph $$G$$ is a set $$N$$ of nodes, a set $$E$$ of edges, a function $$s : E \to N$$ assigning each edge its source, and a function $$t: E \to N$$ assigning each edge its target.

You can visualize the source and target of an edge using this picture:

but remember: it's possible to have $$s(e) = t(e)$$.

If you're a stickler for detail, you may also want to see the precise definition of a "path":

Definition. Given a graph, a path from a node $$x$$ to a node $$y$$ is a finite sequence of edges $$(e_1, \dots, e_n)$$ with

[ s(e_1) = x, \quad t(e_1) = s(e_2), \quad \dots, \quad t(e_{n-1}) = s(e_n), \quad t(e_n) = y. ]

Here $$n = 0, 1, 2, 3, \dots$$ is called the length of the path.

Okay, I think that's enough for today! Next time we'll use this idea to think about databases. For now you should try these exercises, or at least look at other students' solutions:

Exercise 3. Show that $$\textbf{Free}(G)$$ really is a category.

Exercise 4. Completely work out the set of morphisms between each pair of objects in the category called $$\textbf{3}$$, which is

[ \textbf{Free}( [ v_1 \overset{f_1}{\rightarrow} v_2 \overset{f_2}{\rightarrow} v_3 ] ) .]

Exercise 5. The category $$\textbf{3}$$ is an example of a general idea. For each natural number $$n$$ there's a category $$\mathbf{n}$$ that works the same way. Figure out what it is, and figure out the total number of morphisms in this category!

And here's a puzzle for people who love numbers. I'll admit it: I love numbers. But math is not only about numbers, so if you don't love them, don't worry: this puzzle is not important for this course, though it is part of a big subject in math:

Puzzle 105. It's been known since at least the fifth century BC that [ \sqrt 2 = 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots\,}}}}} ] If we cut off this continued fraction we get these rational approximations to $$\sqrt{2}$$: [ \frac11, \frac32, \frac75, \frac{17}{12}, \frac{41}{29}, \frac{99}{70}, \dots] where the denominator of the $$n$$th fraction in this list is called the Pell number $$P_n$$ and the numerator is $$P_n + P_{n-1}$$. Find a graph with two nodes $$x$$ and $$y$$ such that the number of paths of length $$n$$ from $$x$$ to $$y$$ is the $$n$$th Pell number.

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© 2018 John Baez