### Monoidal Closed Categories And Their Deviant Relatives

#### Posted by John Baez

Almost a year ago, Mike Stay and I proudly announced the completion of our Rosetta Stone paper, which explains how symmetric monoidal closed categories show up in physics, topology, logic and computation. But then Theo and Todd helped us spot a serious mistake in our description of a programming language that was supposed to be suitable for work inside any symmetric monoidal closed category — and hence, good for both classical and quantum computation.

We’re almost done fixing that problem now, and Mike is starting to write his thesis on related issues: classical and quantum computation and how they relate to ‘linear’ and also ‘categorified’ versions of the lambda-calculus.

But as a small side-effect, we’ve stumbled upon a deviant definition of ‘monoidal closed category’. I haven’t proved it’s equivalent to the usual definition. I bet it’s not. But I don’t know counterexamples. I wonder if anyone here has thought about such entities before.

Since several experts on monoidal closed categories read this blog, I’m optimistic.

A monoidal category $M$ is closed, or more precisely **right closed**, if for each object $a \in M$, the functor ‘tensoring on the right by $a$’
has a right adjoint.

This right adjoint sends any object $y \in M$ to an object $a \implies y$ of $M$ called the **internal hom**. It’s not always the same as the set of morphisms from $a$ to $y$, which is denoted $hom(a,y)$ It’s kind of like what a guy living inside $M$ would use as a replacement for this set!

Thanks to the definition of right adjoint, we have a natural isomorphism of sets

$hom(x \otimes a, y) \cong hom(x , a \implies y)$

There’s a nice proof that starting with this, we can get a natural isomorphism of objects

$(x \otimes a) \implies y \cong x \implies (a \implies y)$

This is a kind of ‘internal’ version of the previous formula!

Now for my deviant definition. Call a monoidal category $M$ **internally closed** if it’s equipped with a functor

$\implies : M^{op} \times M \to M$

and a natural isomorphism

$(x \otimes a) \implies y \cong x \implies (a \implies y)$

where both sides here are shorthand for functors from $M^{op} \times M^{op} \times M$ to $M$.

Is every internally closed monoidal category actually closed? If not, what are some nice counterexamples? Does anyone think about these things? Is there any beautiful way to add axioms to get a definition equivalent to the usual definition of closed monoidal category?

## Re: Monoidal Closed Categories And Their Deviant Relatives

Let’s see if I understand the question. You have a category $M$, a functor $\otimes: M\times M \to M$ satisfying all the axioms of a monoidal category, and moreover you have a functor $\Rightarrow: M^{op} \times M \to M$ and a natural isomorphism of functors $M^{op} \times M^{op} \times M \to M$:

This would be a right-closed monoidal category if there is a natural isomorphism of functors to SET:

We remark that $M$ is closed-monoidal exactly if we have an isomorphism of functors $M^{op} \times M \to SET$ given by $hom(a,y) \cong hom(1,a\Rightarrow y)$, where $1$ is the monoidal unit.

In any case, let’s break the example. Take $M$ monoidal abelian, and let $a \Rightarrow y = 0$ for any $a, y \in M$, where $0$ is the zero-object in the abelian category, and moreover let’s say that $\Rightarrow$ also kills any morphisms. For that matter, we could pick any constant functor for $\Rightarrow$ (all objects go to our particular choice of object, and all maps go to the identity on that object). Then the natural isomorphism is trivially satisfied — the two sides are in fact _equal_ as functors — and $\Rightarrow$ is not the internal hom of a closed-monoidal category.