nLab premonoidal category

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 $C \Box C \to C$

We might hope that non-strict premonoidal categories are pseudomonoids in $(Cat,\Box)$ seen as a monoidal 2-category, by taking either natural or unnatural transformations as 2-cells. For the requirement that coherence maps are natural, suggests taking natural transformations. However, this is not enough to ensure the coherence maps are central.

Román 22 provides the solution, showing that Freyd categories (or effectful categories), which include premonoidal categories as a special case, are precisely pseudomonoids in a monoidal bicategory of promonads? equipped with their pure tensor product (described in Section 4 of Román 22).

Examples

• Every monoidal category is a premonoidal category.

• If $T$ is a left- and right-strong 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.

Morphisms

A notion of (non-strict) premonoidal functor is somewhat tricky to define. Part of the definition is clear: it should be a functor that preserves the tensor product up to specified coherent central isomorphism. However, the tricky part is whether it should also be required to preserve centrality of morphisms (or even just isomorphisms). Desiderata pulling in opposite directions include:

1. If a premonoidal functor is not required to preserve centrality at least of isomorphisms, then premonoidal functors may not be closed under composition, since we may not be able to define central coherence isomorphisms for $G \circ F$ if $G$ does not preserve the centrality of the coherence isomorphisms for $F$.

2. A morphism $T_1 \to T_2$ of bistrong monads on a monoidal category $C$ induces a functor $C_{T_1} \to C_{T_2}$ which preserves the premonoidal structures strictly, hence clearly up to coherent central isomorphism. However, such a functor does not in general preserve centrality even of isomorphisms; a counterexample can be found in SL13, section 5.2.

It seems unlikely that these desiderata can be reconciled purely in the world of premonoidal categories as usually defined. One solution is to pass to Freyd-categories, which are essentially premonoidal categories equipped with a family of “special” central morphisms forming a cartesian monoidal category with the same tensor product. Morphisms of Freyd-categories are easy to define, and include all the morphisms $C_{T_1} \to C_{T_2}$ if $C$ is cartesian (and if it isn’t, then there should be a suitable non-cartesian generalization of Freyd-categories). Other solutions that use different ways of representing a “special subfamily of central (iso)morphisms” are to consider the tensor product functor of a premonoidal category to be a not-necessarily-saturated anafunctor, or (in homotopy type theory) to allow the underlying category of a premonoidal category to be merely a precategory.

Related pages

• premulticategory
• Freyd category

References

• 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

• Sam Staton and Paul Blain Levy, Universal Properties of Impure Programming Languages. ACM Sigplan Notices, 2013, doi.

• Mario Román, Promonads and String Diagrams for Effectful Categories. Proceedings of Fifth International Conference on Applied Category Theory, 2022, doi.