Duality in Monoidal Categories

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

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, *-autonomous
Or rather, a non-symmetric variant of it called an r-category in [BD13].
categories, and rigid (monoidal) categories. It is well-known that these concepts are all connected in the following way.

  1. Every *-autonomous category is closed monoidal. For all x, y \in \mathcal{C} , the internal-hom [x, y] is given by D^{-1}(Dy \otimes x) , where D is the duality functor.

  2. Every rigid monoidal category is *-autonomous. The internal-hom then simplifies to [x, y] = y \otimes x^* , where {-}^* is the duality functor.

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?

*-autonomy

We’ll start with a positive result for *-autonomy. So the question is this: given a closed monoidal category \mathcal{C} in which the internal-hom is given by tensoring with another object, is this category already *-autonomous?

More formally, is it true that \mathcal{C} is *-autonomous if for all x \in \mathcal{C} , there exists an object Dx \in \mathcal{C} , such that there is an adjunction {-} \otimes x \dashv {-} \otimes Dx?

Almost! In good cases, we can recover what we want from just a little extra condition:

Let \mathcal{C} be a monoidal category. Suppose that for all x \in \mathcal{C} there exist objects Lx, Rx \in \mathcal{C} , such that we have adjunctions {-} \otimes Lx \dashv {-} \otimes x \dashv {-} \otimes Rx. Then \mathcal{C} is *-autonomous.

Using the notion of a *-autonomous category of [BD13]—that is, for every x \in \mathcal{C} the functor \mathcal{C}({-} \otimes x, 1) is representable by Dx —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 \mathcal{C} is rigid monoidal if for all x \in \mathcal{C} we have [x, {-}] \cong {-} \otimes Dx , for some object assignment D \colon \mathrm{Ob}\,\mathcal{C} \longrightarrow \mathrm{Ob}\,\mathcal{C} .

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 ([\mathcal{C}, \mathcal{C}], \circ, \mathrm{Id}) 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 F\colon \mathcal{C} \leftrightarrows \mathcal{C} : \! U with unit \eta \colon \mathrm{Id}_{\mathcal{C}} \Longrightarrow U F and counit \varepsilon \colon F U \Longrightarrow \mathrm{Id}_{\mathcal{C}} . The snake identities for this adjunction look like

Usual snake identities of an adjunction

In particular, we get two such diagrams if we apply everything to the monoidal unit 1 \in \mathcal{C} . Specialised to the adjunction {-} \otimes x \dashv {-} \otimes Dx the above then becomes

Snake identities of an adjunction, specialised to this use-case

These are just the snake identities for duals if we make the definitions \mathrm{ev}_x \mathrel{\vcenter{:}}= \varepsilon_1 and \mathrm{coev}_x \mathrel{\vcenter{:}}= \eta_1 , right? Wrong! In the latter case we, for example, require that (x \otimes \varepsilon_1) \circ (\eta_1 \otimes x) = \mathrm{id}_x. However, the above diagram does not say that! It says that the relation \varepsilon_x \circ (\eta_1 \otimes x) = \mathrm{id}_x holds. This means that we would have to impose the additional conditions that \varepsilon and \eta are morphisms of modules; i.e., \varepsilon_x \overset{\scriptsize{!}}{=} x \otimes \varepsilon_1 = x \otimes \mathrm{coev}_x , 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 \mathcal{D} that is generated by a family of morphisms \eta_{m, n} \colon m \longrightarrow m \otimes n \otimes n \qquad \text{and} \qquad \varepsilon_{m, n} \colon m \otimes n \otimes n \longrightarrow m, and impose relations guaranteeing the naturality of these arrows. There is a subcategory \mathcal{C} of \mathcal{D} in which we additionally require \eta and \varepsilon satisfy the snake equations of an adjunction. One can now show that the category \mathcal{C} is closed monoidal, with the appealing adjunction {-} \otimes n \dashv {-} \otimes n. 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 \mathcal{D} 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 \mathcal{C} , 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!