category with duals (list of them)
dualizable object (what they have)
Recall that a bifunctor to from and (for categories) is simply a functor to from the product category . We can think of this as an operation which is ‘jointly functorial’. But just as a function to from and (for topological spaces) may be continuous in each variable yet not jointly continuous? (continuous from the Tychonoff product ), so an operation between categories can be functorial in each variable separately yet not jointly functorial.
Recall that a monoidal category is a category equipped with a bifunctor (equipped with extra structure such as the associator). Similarly, a premonoidal category is a category equipped with an operation , which is (at least) a function on objects as shown, but one which is functorial only in each variable separately.
A binoidal category is a category equipped with
A morphism in a binoidal category is central if, for every morphism , the diagrams
commute. In this case, we denote the common composites and .
A premonoidal category is a binoidal category equipped with:
such that the following conditions hold.
A strict premonoidal category is a monoidal category in which , , and , and in which , , and are all identity morphisms. (We need the underlying category to be a strict category for this to make sense.)
Similarly, a symmetric premonoidal category is a premonoidal category equipped with a central natural isomorphism (as for , there are two naturality squares unless we use the slick approach), satisfying the usual axioms of a symmetry.
Given categories and functors , a (not necessarily natural) transformation from to consists of, for each object of , a morphism from to in . (So a natural transformation is a transformation that satisfies an extra property.) We can compose transformations using vertical composition (but not horizontal composition).
Given categories , let be the category whose objects are functors from to and whose morphisms are transformations between these functors. This makes into a closed category. We can then define a tensor product by a universal property and make into a monoidal category which is in fact symmetric.
Then a strict premonoidal category is precisely a monoid object in .
It may be possible to weaken the above make a symmetric monoidal 2-category, in which a monoid object is precisely a premonoidal category, but if so, nobody seems to have written this up yet. It is possible, however, to describe part of the structure of a non-strict premonoidal category in terms of . For instance, a binoidal structure on is precisely a functor , and the naturality of the associator can be expressed by saying that it is a natural transformation (with central components) between functors .
Every monoidal category is a premonoidal category.
If is a strong monad on a monoidal category , then the Kleisli category of inherits a premonoidal structure, such that the functor is a strict premonoidal functor. This premonoidal structure is only a monoidal structure if is a commutative monad.
The central morphisms of a premonoidal category form a subcategory , called the centre of , which is a monoidal category. This may define an adjoint functor to the inclusion (I haven't actually checked this).
In the same way that a (strict) monoidal category can be identified with a (strict) 2-category with one object, a strict premonoidal category can be identified with a sesquicategory with one object. In fact, a sesquicategory is precisely a category enriched over the monoidal category described above.