### Monoidal Categories with Projections

#### Posted by Tom Leinster

Monoidal categories are often introduced as an abstraction of categories with products. Instead of having the categorical product $\times$, we have some other product $\otimes$, and it’s required to behave in a somewhat product-like way.

But you could try to abstract *more* of the structure of a category with
products than monoidal categories do. After all, when a category has
products, it also comes with special maps $X \times Y \to X$ and $X \times Y \to Y$
for every $X$ and $Y$ (the projections). Abstracting this leads to
the notion of “monoidal category with projections”.

I’m writing this because over at this thread on magnitude homology, we’re making heavy use of semicartesian monoidal categories. These are simply monoidal categories whose unit object is terminal. But the word “semicartesian” is repellently technical, and you’d be forgiven for believing that any mathematics using “semicartesian” anythings is bound to be going about things the wrong way. Name aside, you might simply think it’s rather ad hoc; the nLab article says it initially sounds like centipede mathematics.

I don’t know whether semicartesian monoidal categories are truly necessary to the development of magnitude homology. But I do know that they’re a more reasonable and less ad hoc concept than they might seem, because:

TheoremA semicartesian monoidal category is the same thing as a monoidal category with projections.

So if you believe that “monoidal category with projections” is a reasonable or natural concept, you’re forced to believe the same about semicartesian monoidal categories.

I’m going to keep this post light and sketchy. A **monoidal category with
projections** is a monoidal category $V = (V, \otimes, I)$ together with a
distinguished pair of maps

$\pi^1_{X, Y} \colon X \otimes Y \to X, \qquad \pi^2_{X, Y} \colon X \otimes Y \to Y$

for each pair of objects $X$ and $Y$. We might call these “projections”. The projections are required to satisfy whatever equations they satisfy when $\otimes$ is categorical product $\times$ and the unit object $I$ is terminal. For instance, if you have three objects $X$, $Y$ and $Z$, then I can think of two ways to build a “projection” map $X \otimes Y \otimes Z \to X$:

think of $X \otimes Y \otimes Z$ as $X \otimes (Y \otimes Z)$ and take $\pi^1_{X, Y \otimes Z}$; or

think of $X \otimes Y \otimes Z$ as $(X \otimes Y) \otimes Z$, use $\pi^1_{X \otimes Y, Z}$ to project down to $X \otimes Y$, then use $\pi^1_{X, Y}$ to project from there to $X$.

One of the axioms for a monoidal category with projections is that these two maps are equal. You can guess the others.

A monoidal category is said to be **cartesian** if its monoidal structure
is given by the categorical (“cartesian”) product. So, any cartesian monoidal
category becomes a monoidal category with projections in an obvious way:
take the projections $\pi^i_{X, Y}$ to be the usual product-projections.

That’s the motivating example of a monoidal category with projections, but there are others. For instance, take the ordered set $(\mathbb{N}, \geq)$, and view it as a category in the usual way but with a reversal of direction: there’s one object for each natural number $n$, and there’s a map $n \to m$ iff $n \geq m$. It’s monoidal under addition, with $0$ as the unit. Since $m + n \geq m$ and $m + n \geq n$ for all $m$ and $n$, we have maps $m + n \to m$ and $m + n \to n$.

In this way, $(\mathbb{N}, \geq)$ is a monoidal category with projections. But it’s not cartesian, since the categorical product of $m$ and $n$ in $(\mathbb{N}, \geq)$ is $max\{m, n\}$, not $m + n$.

Now, a monoidal category $(V, \otimes, I)$ is **semicartesian** if the unit
object $I$ is terminal. Again, any cartesian monoidal category gives an
example, but this isn’t the only kind of example. And again, the ordered set
$(\mathbb{N}, \geq)$ demonstrates this: with the monoidal structure just
described, $0$ is the unit object, and it’s terminal.

The point of this post is:

TheoremA semicartesian monoidal category is the same thing as a monoidal category with projections.

I’ll state it no more precisely than that. I don’t know who this result is due to; the nLab page on semicartesian monoidal categories suggests it might be Eilenberg and Kelly, but I learned it from a Part III problem sheet of Peter Johnstone.

The proof goes roughly like this.

Start with a semicartesian monoidal category $V$. To build a monoidal category with projections, we have to define, for each $X$ and $Y$, a projection map $X \otimes Y \to X$ (and similarly for $Y$). Now, since $I$ is terminal, we have a unique map $Y \to I$. Tensoring with $X$ gives a map $X \otimes Y \to Y \otimes I$. But $Y \otimes I \cong Y$, so we’re done. That is, $\pi^1_{X, Y}$ is the composite

$X \otimes Y \stackrel{X \otimes !}{\longrightarrow} X \otimes I \cong X.$

After a few checks, we see that this makes $V$ into a monoidal category with projections.

In the other direction, start with a monoidal category $V$ with projections. We need to show that $V$ is semicartesian. In other words, we have to prove that for each object $X$, there is exactly one map $X \to I$. There’s at least one, because we have

$X \cong X \otimes I \stackrel{\pi^2_{X, I}}{\longrightarrow} I.$

I’ll skip the proof that there’s at most one, but it uses the axiom that the projections are natural transformations. (I didn’t mention that axiom, but of course it’s there.)

So we now have a way of turning a semicartesian monoidal category into a monoidal category with projections and vice versa. To finish the proof of the theorem, we have to show that these two processes are mutually inverse. That’s straightforward.

Here’s something funny about all this. A monoidal category with
projections appears to be a monoidal category with extra *structure*, whereas
a semicartesian monoidal category is a monoidal category with a certain
*property*. The theorem tells us that in fact, there’s at most one possible
way to equip a monoidal category with projections (and there *is* a way if
and only if $I$ is terminal). So having projections turns out to be a
property, not structure.

And that is my defence of semicartesian monoidal categories.

## Re: Monoidal Categories with Projections

Typo: “centipede mathematics” links to the nLab page on semicartesian monoidal categories.