# Warming Up to Enriched Category Theory, Part 1

*It's no secret that I like category theory.* It's a common theme on this blog, and it provides a nice lens through which to view old ideas in new ways — and to view *new *ideas in new ways! Speaking of new ideas, my collaborators and I are planning to upload a new paper on the arXiv soon. I've really enjoyed the work and can't wait to share it with you. But first, you'll have to know a little something about **enriched category theory**. (And before *that*, you'll have to know something about ordinary category theory... here's an intro!) So that's what I'd like to introduce today.

It's a warm up, if you will.

## What is enriched category theory?

As the name suggests, it's like a "richer" version of category theory, and it all starts with a simple observation. * (Get your category theory hats on, people. We're jumping right in!)*

In a category, you have some objects and some arrows between them, thought of as relationships between those objects. Now in the formal definition of a category, we usually ask for a *set's worth* of morphisms between any two objects, say $X$ and $Y$. You'll typically hear something like, "The hom set $\text{hom}(X,Y)$ bla bla...."

Now here's the thing. Quite often in mathematics, the set $\text{hom}(X,Y)$ may not just be a set. It could, for instance, be a set equipped with extra structure. You already know lots of examples. Let's think about about linear algebra, for a moment.

If we have a pair of real vector spaces, say $V$ and $W$, then the set of linear transformations $V\to W$ has lots of structure: if $f,g\colon V\to W$ are linear transformations, then their sum $f+g$ is also a linear transformation from $V$ to $W$, and so is the scalar multiple $kf$ for any real number $k$. In fact, the point here is that the set $\hom(V,W)$ of linear transformations is *itself* a real vector space. So the hom sets evidently have "richer" structure. They are *enriched! *Now, linear algebra is just one example. You can think of others! The set of continuous functions between a pair of topological spaces can *itself* be given a topology; the set of homomorphisms between a pair of abelian groups is *itself *an abelian group; and so on.

And *that's *the main idea behind enriched category theory. (Well, that's the gist. The theory gets deep quickly.)

You have a category $\mathsf{C}$, and the hom *sets* between the objects in $\mathsf{C}$ are *themselves* objects in some other category, which is often called the **base category** or the category over which $\mathsf{C}$ is **enriched**. In the examples above, the categories were each enriched over *themselves*, but it's totally fine to have a category $\mathsf{C}$ enriched over a *different* category.

Admittedly, the graphic above isn't the whole story. For the math to *really *work out, we have to be a little careful with the axioms for composition of morphisms and identity morphisms, and so there's a little more to say. Indeed, there is a very *formal* definition of an enriched category, but I'm not sharing it just yet. We're still warming up! In fact, I want to dial things back even further and consider the following super simple scenario.

## Let's take it down a notch.

Suppose we have a pair of objects $X$ and $Y$, and let's further suppose there is *at most one* morphism from $X$ to $Y.$ This is much simpler than our examples above from linear algebra, group theory, and topology, where in principle there could've been *loads *of morphisms. But we want to keep things simple for now.

So, suppose we're in a situation where either there's an arrow $X\to Y$ or there's not.

Now, let's *also *consider the possibility that that arrow can be "decorated" with — or simply *replaced* by — some number. Perhaps that number indicates the degree to which that arrow is there. Or perhaps it represents the amount of effort it takes to "get" from $X$ to $Y$. Or maybe it represents the *probability (*or some fuzziness*) of going from $X$ to $Y$. Or the *distance *it takes to travel from $X$ to $Y$. Or maybe it just represents the Boolean *truth-value* of whether or not that arrow is even there. Use your imagination!

Imagination is good, but so is thinking systematically. So let's rope things in a bit. I really like those last three suggestions — truth values, distances, and probabilities — and I'd like us to be little more formal about it. To that end, let's say the arrow $X\to Y$ can be either be decorated with a number from the two-element set $\{0,1\}$ (thought of as truth values), or perhaps the unit interval $[0,1]$ (thought of as probabilities or some fuzziness), or perhaps the set of nonnegative extended reals $[0,\infty]$ (thought of as distances).

See the analogy, here? Earlier, the collection of arrows from $X$ to $Y$ was an object in the category $\mathsf{Vect}$ of vector spaces, or the category $\mathsf{Top}$ or topological spaces, or the category $\mathsf{AbGrp}$ of abelian groups, or.... *But now *the "collection"* *consisting of the single arrow $X\to Y$ is basically just an element in the set $\{0,1\}$, or the set $[0,1]$, or the set $[0,\infty]$, or...

See where we're going with this? I'll summarize it as a question and answer:

**Question**: *Is there a way to view the sets $\{0,1\}$ and $[0,1]$ and $[0,\infty]$ as categories so that the number "$\text{hom}(X,Y)$" is actually an object in that category? And is there a more formal way to talk about categories "enriched" over these categories?*

**Answer:** YES and YES!

And rather than belaboring this point any longer, let's cut to the chase.

## Ramping back up.

Each of the three sets above are examples of **preordered sets,** which are very easy-to-understand kinds of categories. A preordered set, or simply a *preorder*, is a set equipped with a reflexive and transitive relation typically denoted by $\leq$. If the relation is also antisymmetric, then it's actually a *partially-ordered set*, i.e. a poset. But it turns out that just having a preorder means you automatically have yourself a category. The objects are elements in the set, and morphisms are provided by $\leq$. Identity morphisms are provided by reflexivity, and composition is provided by transitivity.

So, every preorder is a category.

In particular, there is a category $\{0\leq 1\}$ of **truth values, **whose only two objects are the numbers $0$ and $1$ and where the only non-identity morphism is $0\leq 1$. The **unit interval** $[0,1]$ is also a category, since it's a preorder with the usual ordering $\leq$. You can think of having a morphism $0.3\to 0.75$ since $0.3\leq 0.75$. And the **nonnegative extended reals** $[0,\infty]$ are also a preorder, but for historical reasons (i.e. Lawvere — we'll get to him later!), we'll view it as a category where there's an arrow between real numbers $a\to b$ whenever $a\geq b$, which is the *opposite* of the usual ordering.

Now a word of caution: *don't let the simplicity of these preorders fool you.* Categories *enriched over* them pave the way for *tons* of nice examples, and lovely theory, and current threads of research.

But wait — *what exactly IS a category enriched over a preorder?*

I'll tell you next time.

There's so much more to say, but this post is already quite long.

Stay tuned!

*If you're already an expert on enriched category theory, then you'll know that when folks enrich over $[0,1]$, they're usually thinking about fuzzy logic. And you'll *also* know that fuzzy logicians are pretty adamant that you *not *think of elements in $[0,1]$ as probabilities. But let's allow ourselves to be flexible about this.