It took us two posts, but we showed that the category of representations of a Hopf algebra has duals. This is on top of our earlier result that the category of representations of any bialgebra is monoidal. Let’s look at this a little more conceptually.
Earlier, we said that a bialgebra is a comonoid object in the category of algebras over . But let’s consider this category itself. We also said that an algebra is a category enriched over , but with only one object. So we should really be thinking about the category of algebras as a full sub-2-category of the 2-category of categories enriched over .
So what’s a comonoid object in this 2-category? When we defined comonoid objects we used a model category . Now let’s augment it to a 2-category in the easiest way possible: just add an identity 2-morphism to every morphism!
But the 2-category language gives us a bit more flexibility. Instead of demanding that the morphism satisfy the associative law on the nose, we can add a “coassociator” 2-morphism to our model 2-category. Similarly, we dispense with the left and right counit laws and add left and right counit 2-morphisms. Then we insist that these 2-morphisms satisfy pentagon and triangle identities dual to those we defined when we talked about monoidal categories.
What we’ve built up here is a model 2-category for weak comonoid objects in a 2-category. Then any weak comonoid object is given by a 2-functor from this 2-category to the appropriate target 2-category. Similarly we can define a weak monoid object as a 2-functor from the opposite model 2-category to an appropriate target 2-category.
So, getting a little closer to Earth, we have in hand a comonoid object in the 2-category of categories enriched over — our algebra . But remember that a 2-category is just a category enriched over categories. That is, between (considered as a category) and we have a hom-category . The entry in the first slot is described by a 2-functor from the model category of weak comonoid objects to the 2-category of categories enriched over . This hom-functor is contravariant in the first slot (like all hom-functors), and so the result is described by a 2-functor from the opposite of our model 2-category. That is, it’s a weak monoid object in the 2-category of all categories. And this is just a monoidal category!
This is yet another example of the way that hom objects inherit structure from their second variable, and inherit opposite structure from their first variable. I’ll leave it to you to verify that a monoidal category with duals is similarly a weak group object in the 2-category of categories, and that this is why a Hopf algebra — a (weak) cogroup object in the 2-category of categories enriched over has dual representations.