Idea

A promonoidal category is like a monoidal category in whose structure (namely, tensor product and unit object) we have replaced functors by profunctors.

Definition

A promonoidal category is a pseudomonoid in the monoidal bicategory Prof. This means that it is a category $A$ together with

• A profunctor $P \colon A\times A ⇸ A$.
• A profunctor $J\colon 1$ ⇸ $A$.
• Associativity and unit isomorphisms $P \odot (P\times 1) \cong P\odot (1\times P)$, $P\odot (J\times 1) \cong 1$, and $P\odot (1\times J) \cong 1$.
• The usual pentagon and unit conditions hold, as in a monoidal category.

Recalling that a profunctor $A$ ⇸ $B$ is defined to be a functor of the form $B^{op}\times A \to Set$, we can make this more explicit. We can also generalize it by replacing Set by a Benabou cosmos $V$ and $A$ by a $V$-enriched category; then a profunctor is a $V$-enriched functor $B^{op}\times A \to V$.

Thus, we obtain the following as an explicit definition of promonoidal $V$-category:

We have the following data

1. A $V$-enriched category $A$.

2. A $3$-ary enriched functor $P:A^\op \otimes A \otimes A\to V$. For notational clarity, we may write $P(a,b,c)$ as $P(a,b \diamond c)$.

3. A $V$-functor $J:A^{op}\to V$.

and enriched natural isomorphisms

1. $\lambda_{ab}:\int^x (J(x) \otimes P(b,a \diamond x))\to A(b,a)$

2. $\rho_{ab}: \int^x ( J(x)\otimes P(b,x \diamond a))\to A(b,a)$

3. $\alpha_{abcd}: \int^x (P(x,a\diamond b)\otimes P(d,x\diamond c)) \to \int^x(P(x,b\diamond c)\otimes P(d,a\diamond x))$

satisfying the pentagon and unit axioms for promonoidal categories. Explicitly, writting $P^{A}_{B,C}$ for $P(A,B;C)$, $\mathsf{h}^{A}_{B}$ for $\mathrm{Hom}_{A}(A,B)$, $J^X$ for $J(X)$, and $\diamond$ for composition of profunctors, we require the following conditions to hold:

1. The triangle identity for promonoidal categories. For each $A,B,C\in\mathrm{Obj}(A)$, the diagram

   \begin{imagefromfile} “file_name”: “pro-triangle-corrected.svg” \end{imagefromfile}

   commutes;

2. The pentagon identity for promonoidal categories. For each $A,B,C,D,E\in\mathrm{Obj}(A)$, the diagram

   \begin{imagefromfile} “file_name”: “pro-pentagon.svg” \end{imagefromfile}

   commutes.

Properties

Versus monoidal categories

Since any functor induces a representable profunctor, any monoidal category can be regarded as a promonoidal category. A given promonoidal category arises in this way if and only if the profunctors $P$ and $J$ are representable.

Day convolution

A promonoidal structure on $A$ suffices to induce a monoidal structure on $V^{A^{op}}$ by Day convolution. In fact, given a small $V$-category $A$, there is an equivalence of categories between

1. the category of pro-monoidal structures on $A$, with strong pro-monoidal functors between them, and

2. the category of biclosed monoidal structures on $V^{A^{op}}$, with strong monoidal functors between them.

Versus multicategories

A promonoidal structure on $A$ can be identified with a particular sort of multicategory structure on $A^{op}$, i.e. with a co-multicategory structure on $A$. The set $P(x, y, z)$ is regarded as the set of co-multimorphisms $x \to (y,z)$.

More generally, we define a co-multicategory $\bar A$ as follows. The objects of $\bar A$ are the objects of $A$. The co-multimorphisms $b\to a_1\dots a_n$ in $\bar A$ are defined by induction on $n$ as follows: $\bar A(b;)=Jb$, and $\bar A(b;a_1,\dots,a_{n+1})=\int^x\bar A(x;a_1,\dots,a_n)\otimes P(b,x\diamond a_{n+1})$.

Not every co-multicategory arises from a promonoidal one in this way. Roughly, a promonoidal category is a co-multicategory whose $n$-ary co-multimorphisms are determined by the binary, unary, and nullary morphisms. In general, co-multicategories can be identified with a certain sort of “lax promonoidal category”.

In fact, promonoidal categories correspond exactly to the exponentiable? multicategories: see multicategory for more information.

Notes

Brian Day introduced the notion of a “premonoidal” category in (Day 1970), and later renamed this to a “promonoidal” category in (Day 1974) while reformulating the identity and associativity isomorphisms $\lambda,\rho,\alpha$ explicitly in terms of profunctor composition. However, note that his definition is op’d from the definition used in this article, in the sense that a Day-promonoidal structure on a category $C$ corresponds to a pseudomonoid structure on $C^{op}$ in Prof. In particular, one example Day considers is that of a closed category, which is actually a co-promonoidal category in the sense used here (analogous to the co-promonoidal structure on a multicategory described above).

Regarding monoidal categories as promonoidal is useful in order to express extra structure on them, such as closedness, $\ast$-autonomy, or compact closedness, in abstract bicategorical terms: these notions can be defined by adding extra structure to a pseudomonoid in the monoidal bicategory Prof (i.e. a promonoidal category), but the extra structure does not lie inside the sub-monoidal bicategory Cat.

Related pages

• monoidal category, multicategory
• Day convolution
• A (co-)promonoidal poset is called a ternary frame, when its Day convolution monoidal category is used to model substructural logic.

References

• Brian Day, On closed categories of functors, Lecture Notes in Mathematics 137 (1970), 1-38.

• Brian Day, An embedding theorem for closed categories, Lecture Notes in Mathematics 420 (1974), 55-64.

• Day, Panchadcharam and Street, On centres and lax centres for promonoidal categories.

The relationship between multicategories, promonoidal categories, lax monoidal categories, and monoidal categories is exposited in:

• Brian Day and Ross Street, Lax monoids, pseudo-operads, and convolution, Contemporary Mathematics 318 (2003): 75-96.