# nLab premonoidal category

Premonoidal categories

### Context

#### Monoidal categories

monoidal categories

# Premonoidal categories

## Idea

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.

## Definition

A binoidal category is a category $C$ equipped with

• for each pair $x,y$ of objects of $C$, an object $x \otimes y$;
• for each object $x$ a functor $x \rtimes -$ whose action on objects sends $y$ to $x \otimes y$
• for each object $x$ a functor $- \ltimes x$ whose action on objects sends $y$ to $y \otimes x$

A morphism $f\colon x \to y$ in a binoidal category is central if, for every morphism $f'\colon x' \to y'$, the diagrams

$\array { x \otimes x' & \overset{x \rtimes f'}\to & x \otimes y' \\ \mathllap{f \ltimes x'}\downarrow & & \downarrow\mathrlap{f \ltimes y'} \\ y \otimes x' & \underset{y \rtimes f'}\to & y \otimes y' \\ }$

and

$\array { x' \otimes x & \overset{x' \rtimes f}\to & x' \otimes y \\ \mathllap{f' \ltimes x}\downarrow & & \downarrow\mathrlap{f' \ltimes y} \\ y' \otimes x & \underset{y' \rtimes f}\to & y' \otimes y \\ }$

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:

• an object $I$;
• for each triple $x,y,z$ of objects, a central isomorphism $\alpha_{x,y,z}\colon (x \otimes y) \otimes z \to x \otimes (y \otimes z)$; and
• for each object $x$, central isomorphisms $\lambda_x\colon x \otimes I \to x$ and $\rho_x\colon I \otimes x \to x$;

such that the following conditions hold.

• all possible naturality squares for $\alpha$, $\lambda$, and $\rho$ (which make sense since we have central morphisms) commute. Note that when written out explicitly in terms of the functors $x\rtimes -$ and $-\ltimes x$, we need three different naturality squares for $\alpha$. (But it is possible to rephrase $\alpha$ as a single natural transformation using the slick version below.)
• the pentagon law holds for $\alpha$, as in a monoidal category.
• the triangle law holds for $\alpha$, $\lambda$, and $\rho$, as in a monoidal category.

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.

## Slick version

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 $C\Box C\Box C \to C$.

## Examples

• 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.

## Properties

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).

PostScript

• Alan Jeffrey, Premonoidal categories and a graphical view of programs, pdf file