# nLab promonoidal category

Promonoidal categories

### Context

#### Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

#### Enriched category theory

enriched category theory

# Promonoidal categories

## 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 $J:A^{op}\to V$.

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

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

## 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.

## 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.

Last revised on March 4, 2024 at 13:46:52. See the history of this page for a list of all contributions to it.