Let's jump right in to where we left off in part 1 of our warm-up to enriched category theory. If you'll recall from last time, we saw that the set of truth values $\{0, 1\}$ and the unit interval $[0,1]$ and the nonnegative extended reals $[0,\infty]$ were not just sets but actually preorders and hence categories. We also hinted at the idea that a "category enriched over" one of these preorders (whatever that means — we hadn't defined it yet!) looks something like a collection of objects $X,Y,\ldots$ where there is at most one arrow between any pair $X$ and $Y$, and where that arrow can further be "decorated with" —or simply replaced by — a number from one of those three exemplary preorders.

With that background in mind, my goal in today's article is to say exactly what a category enriched over a preorder is. The formal definition — and the intuition behind it — will then pave the way for the notion of a category enriched over an arbitrary (and sufficiently nice) category, not just a preorder.

En route to this goal, it will help to make a couple of opening remarks.

Two things to think about.

First, take a closer look at the picture on the right. I've written "$\text{hom}(X,Y)$" in quotation marks because the notation $\text{hom}(-,-)$ is often used for a set of morphisms in ordinary category theory. But the  point of this discussion is that we're not just interested in sets! So we should use better notation: let's refer to the number associated to a pair of objects $XY$ and $Y$ as $\mathcal{C}(X,Y)$, where the letter "$\mathcal{C}$" reminds us there's an (enriched) $\mathcal{C}$ategory being investigated.

Second, for the theory to work out nicely, it turns out that preorders need a little more added to them.

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 coauthors 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.

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...."
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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.

Over the years, the articles on this blog have spanned a wide range of audiences, from fun facts (Multiplying Non-Numbers), to undergraduate level (The First Isomorphism Theorem, Intuitively), to graduate level (What is an Operad?), to research level. Today's article is more on the fun-fact side of things, along with—like most articles here—an eye towards category theory.

So here's a fun fact about greatest common divisors (GCDs) and the Fibonacci sequence $F_1,F_2,F_3,\ldots$, where $F_1=F_2=1$ and $F_n:=F_{n-1} + F_{n-2}$ for $n>1$. For all $n,m\geq 1$,

In words, the greatest common divisor of the $n$th and $m$th Fibonacci numbers is the Fibonacci number whose index is the greatest common divisor of $n$ and $m$. (Here's a proof.) Upon seeing this, your "spidey senses" might be tingling. Surely there's some structure-preserving map $F$ lurking in the background, and this identity means it has a certain nice property. But what is that map? And what structure does it preserve? And what's the formal way to describe the nice property it has?

The short answer is that the natural numbers $\mathbb{N}=\{1,2,3,\ldots\}$ form a partially ordered set (poset) under division, and the function $F\colon \mathbb{N}\to\mathbb{N}$ defined by $n\mapsto F_n:=F(n)$ preserves meets: $F_n\wedge F_m = F(n\wedge m)$.

Quantum entanglement is, as you know, a phrase that's jam-packed with meaning in physics. But what you might not know is that the linear algebra behind it is quite simple. If you're familiar with singular value decomposition (SVD), then you're 99% there. My goal for this post is to close that 1% gap. In particular, I'd like to explain something called the Schmidt rank in the hopes of helping the math of entanglement feel a little less... tangly. And to do so, I'll ask that you momentarily forget about the previous sentences. Temporarily ignore the title of this article. Forget we're having a discussion about entanglement. Forget I mentioned that word. And let's start over. Let's just chat math.

Let's talk about SVD.

Singular Value Decomposition

SVD is arguably one of the most important, well-known tools in linear algebra. You are likely already very familiar with it, but here's a lightening-fast recap. Every matrix $M$ can be factored as $M=UDV^\dagger$ as shown below, called the singular value decomposition of $M$. The entries of the diagonal matrix $D$ are nonnegative numbers called singular values, and the number of them is equal to the rank of $M$, say $k$. What's more, $U$ and $V$ have exactly $k$ columns, called the left and right singular vectors, respectively.

There are different ways to think about this, depending on which applications you have in mind. I like to think of singular vectors as encoding meaningful "concepts" inherent to $M$, and of singular values as indicating how important those concepts are.

This is the official launch week of our new book, Topology: A Categorical Approach, which is now available for purchase! We're also happy to offer a free open access version through MIT Press at topology.mitpress.mit.edu.

Inside, you'll find a presentation of basic, point-set topology from the perspective of category theory, targeted at graduate students in a first-semester course on topology. The idea is that most of these students are already somewhat familiar with the point-set ideas through a course on analysis or undergraduate topology. For this reason, many graduate-level instructors are tempted to rush through point-set topology or to skip it altogether to reach algebraic topology, which can be more fun to learn and to teach.

Our book presents an alternative to this approach. Rather than skipping over the basic ideas, we view this as an excellent opportunity to introduce students to the modern viewpoint of category theory.