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Graphs, language, and categories

written by Eric J. Ma on 2020-12-18 | tags: math graph theory category theory programming

Just as I began vacation, my brain started going into overdrive. (This is a quirk of my head.) One of the epiphanies I arrived at involved graphs, programming languages, and categories.

Let's talk about multiple dispatch

Multiple dispatch is a programming construct. It essentially allows us to write functions that:

  1. Share the same name, but
  2. Exhibit different behaviour based on the types of the arguments passed in.

This idea is baked into the Julia programming language and is part of what makes Julia so composable. Python programmers don't have this luxury, partially because multiple dispatch isn't part of the language design and partially because of namespaces. That said, Matt Rocklin and a few others maintain the multipledispatch Python package, which implements multiple dispatch in a pretty sane fashion. To borrow the example from the package, we might design a function called add, but need it to operate differently depending on whether we put in integers or Python objects:

@dispatch(int, int)
def add(a, b):
    """Adds two numbers together."""
    return a + b

@dispatch(object, object)
def add(a, b):
    """Performs a concatenation between two strings."""
    return f"{a}{b}"

Is this a good idea? I used to wonder when I'd know enough to answer the question. I think I do now. But first, we have to take a detour to graphs.

Let's talk about graphs

Graphs are a data structure that allows us to model entities and their relationships. Entities are nodes, while relationships are edges. I usually teach a tutorial at PyCon, SciPy and ODSC titled "Network Analysis Made Simple", in which we use NetworkX as a tool to learn applied graph theory. It's been running since 2014, and I've learned a lot, made a lot of friends, and had new opportunities spring up through teaching it.

As a data model, graphs are handy for reasoning because for our logic to make sense, we must be extremely clear on our definitions. What exactly constitutes a node? What exactly constitutes a relationship? What connections are allowed to exist between nodes (or, surprise! categories of nodes)? These answers must be precise.

This then brings us to categories.

Let's talk about categories

I think it was from Eric Schles that I learned that there was this entire field of mathematics called "category theory". (This was likely ODSC 2017, while both of us were chatting in the speaker's lounge.) Also, I started reading the blog of Tai Danae-Bradley, who trained as a category theorist at NYU and even has a book on it. From one of her excellent blog posts, I learned the core of category theory is about: "...collections of objects that can relate to each other via morphisms..."

Sounds familiar... it sounds like a graph!

So... what about multiple dispatch?

If you stepped back one moment and looked carefully at the add example, you'll notice that we have a microcosm of category theory being brought to life in a programming language.

There's a category of objects called objects, and there's a (sub-)category of objects called integers. There are relations between integers called add, and a similar relation between objects, also called add. Both have a relationship named add, but each add differs because the "integers" case is a particular case that needs to be handled differently from the "objects" case.

You could construct a graph to represent it all, one that looks like this:

(int, int) --add--> int
(obj, obj) --add--> str

If we think about it carefully, because integers are a sub-category of objects, the second case (obj, obj) covers the child cross-type cases.

So what gives, what's the epiphany?

Multiple dispatch works because linguistically, we might need to reuse the same category of relations, as indicated by their shared name, for different object categories, for no other reason than doing so is the semantically correct thing to do for our problem. And doing so allows us to leverage composition to write cleaner and interoperable programs.

Cite this blog post:
    author = {Eric J. Ma},
    title = {Graphs, language, and categories},
    year = {2020},
    month = {12},
    day = {18},
    howpublished = {\url{}},
    journal = {Eric J. Ma's Blog},
    url = {},

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