category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
A premonoidal category is a generalisation of a monoidal category, applied by John Power and his collaborators to denotational semantics in computer science. There, the Kleisli category of a strong monad provides a model of call-by-value? programming languages. In general, if the original category is monoidal, the Kleisli category will only be premonoidal.
Recall that a bifunctor from $C$ and $D$ to $E$ (for $C,D,E$ categories) is simply a functor to $E$ from the product category $C \times D$. We can think of this as an operation which is ‘jointly functorial’. But just as a function to $X$ from $Y$ and $Z$ (for $X,Y,Z$ topological spaces) may be continuous in each variable yet not jointly continuous? (continuous from the Tychonoff product $Y \times Z$), so an operation between categories can be functorial in each variable separately yet not jointly functorial.
Recall that a monoidal category is a category $C$ equipped with a bifunctor $C \times C \to C$ (equipped with extra structure such as the associator). Similarly, a premonoidal category is a category equipped with an operation $C \times C \to C$, 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 $C$ equipped with
A morphism $f\colon x \to y$ in a binoidal category is central if, for every morphism $f'\colon x' \to y'$, the diagrams
and
commute. In this case, we denote the common composites $f \otimes f'\colon x \otimes x' \to y \otimes y'$ and $f' \otimes f\colon x' \otimes x \to y' \otimes y$.
A premonoidal category is a binoidal category equipped with:
such that the following conditions hold.
A strict premonoidal category is a premonoidal category in which $(x \otimes y) \otimes z = x \otimes (y \otimes z)$, $x \otimes I = x$, and $I \otimes x = x$, and in which $\alpha_{x,y,z}$, $\lambda_x$, and $\rho_x$ are all identity morphisms. (We need the underlying category $C$ 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 $x\otimes y \cong y\otimes x$ (as for $\alpha$, there are two naturality squares unless we use the slick approach), satisfying the usual axioms of a symmetry.
As a strict monoidal category is a monoid in the cartesian monoidal category Cat, so a strict premonoidal category is a monoid in the symmetric monoidal category $(Cat,\Box)$, where $\Box$ is the funny tensor product.
From this point of view, a binoidal category is just a category $C$ with a functor $C \Box C \to C$
It may be possible to make $(Cat,\Box)$ a symmetric monoidal 2-category, in which a pseudomonoid object is precisely a non-strict 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 $(Cat,\Box)$. For instance, a binoidal structure on $C$ is precisely a functor $C\Box C \to C$, and the naturality of the associator $\alpha$ can be expressed by saying that it is a natural transformation (with central components) between functors $C\Box C\Box C \to C$.
Every monoidal category is a premonoidal category.
If $T$ is a strong and costrong monad on a monoidal category $C$ (e.g. a strong monad on a braided monoidal category), then the Kleisli category $C_T$ of $T$ inherits a premonoidal structure, such that the functor $C\to C_T$ is a strict premonoidal functor. This premonoidal structure is only a monoidal structure if $T$ is a commutative monad.
A strict premonoidal category is the same as a sesquicategory with one object, so any object of a sesquicategory has a corresponding premonoidal category whose objects are endomorphisms and arrows are 2-cells.
The central morphisms of a premonoidal category $C$ form a subcategory $Z(C)$, called the centre of $C$, which is a monoidal category. This defines a right adjoint functor to the inclusion $MonCat \hookrightarrow PreMonCat$ using the definition of functor of premonoidal categories in Power-Robinson 97.
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 $(Cat,\otimes)$ described above.
John Power and Edmund Robinson, Premonoidal categories and notions of computation, Math. Structures Comput. Sci., 7(5):453–468, 1997. Logic, domains, and programming languages (Darmstadt, 1995).
Alan Jeffrey, Premonoidal categories and a graphical view of programs, pdf file
Last revised on May 22, 2020 at 13:08:23. See the history of this page for a list of all contributions to it.