Posted on 2023-01-10 · last modified: 2023-01-12 · 4 min read · maths

Contents

I have a new preprint on the arXiv! It is joint work with
Sebastian Halbig, and concerns itself with the interplay of different
structures on monoidal categories that give rise to a notion of
“duality”. At five pages, it is a very short paper; yet I’d still like
to give a little teaser as to what kind of question we sought to answer.

## Setting the scene¶

We mainly concerned ourselves with three notions of*duality*for (non-symmetric!) monoidal categories: closed monoidal categories, *-autonomousOr rather, a non-symmetric variant of it called an

categories, and rigid (monoidal) categories. It
is well-known that these concepts are all connected in the following
way.
*r-category*in [BD13].An obvious next question one could ask is: does this already characterise rigid and *-autonomous categories? More explicitly, are there any conditions one could impose on the internal-hom, such that closedness already implies rigidity? What about *-autonomy?

- Every *-autonomous category is closed monoidal. For all , the internal-hom is given by , where is the duality functor.
- Every rigid monoidal category is *-autonomous. The internal-hom then simplifies to , where is the duality functor.

## *-autonomy¶

We’ll start with a positive result for *-autonomy. So the question is this: given a closed monoidal category in which the internal-hom is given by tensoring with another object, is this category already *-autonomous? More formally, is it true that is *-autonomous if for all , there exists an object , such that there is an adjunction Almost! In good cases, we can recover what we want from just a little extra condition:Let be a monoidal category. Suppose that for all there exist objects , such that we have adjunctions Then is *-autonomous.Using the notion of a *-autonomous category of [BD13]—that is, for every the functor is representable by —this becomes an exercise in “Yoneda Yoga”. More precisely, one uses the fact that the Yoneda embedding is fully faithful a lot. Try it yourself!

## Rigidity¶

At first sight, it’s not even clear there is anything to show for rigidity. Something one is immediately tempted to do is to conjecture the following:A closed monoidal category is rigid monoidal if for all we have , for some object assignment .This seems sensible; after all, the snake identities of an adjunction look almost completely the same as the ones for a dual!

This is not a coincidence; for example, adjoints in the monoidal
category are
exactly duals!.

However, if
one sits down and actually writes down the diagrams, something doesn’t
quite fit. As a reminder, suppose we have an adjunction
with unit
and counit
.
The snake identities for this adjunction look like
In particular, we get two such diagrams if we apply everything to the
monoidal unit . Specialised to the adjunction
the above then becomes
These are just the snake identities for duals if we make the definitions
and , right?
Wrong! In the latter case we, for example, require that
However, the above diagram does *not*say that! It says that the relation holds. This means that we would have to impose the additional conditions that and are morphisms of modules; i.e., , as well as a dual statement. This is not the case in general. Finding a counterexample now works by exploiting exactly this fact: we write down a syntactic category that is generated by a family of morphisms and impose relations guaranteeing the naturality of these arrows. There is a subcategory of in which we additionally require and satisfy the snake equations of an adjunction. One can now show that the category is closed monoidal, with the appealing adjunction However, it is not rigid! The proof exploits certain strong monoidal functors to the category of finite-dimensional vector spaces, and shows that the subset of arrows*in*that contains one of the snake identities for duals is (i) closed under exactly these relations, and (ii) all morphisms in this set have length at least two. Hence, if we project any morphism down to , it can’t possibly be the identity, and thus the snake identities for duals do not hold. If you want more details, check the paper!