nLab
promonoidal category

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 $P:A\times A⇸A$.
• A profunctor $J:1⇸A$.
• 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 $B^{\mathrm{op}}\times A\to \mathrm{Set}$, we can make this more explicit. We can also generalize it by replacing $\mathrm{Set}$ by a Benabou cosmos $V$ and $A$ by a $V$-enriched category; then a profunctor is a $V$-functor $B^{\mathrm{op}}\times A\to V$.

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

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

2. A $3$-ary 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 $J:A^{\mathrm{op}}\to V$.

and natural isomorphisms

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

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

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

satisfying the pentagon and unit axioms.

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 $V^{A^{\mathrm{op}}}$ 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 $V^{A^{\mathrm{op}}}$, with strong monoidal functors between them.

Versus multicategories

A promonoidal structure on $A$ can be identified with a particular sort of multicategory structure on $A^{\mathrm{op}}$, 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 $\overline{A}$ as follows. The objects of $\overline{A}$ are the objects of $A$. The co-multimorphisms $b\to a_1\dots a_n$ in $\overline{A}$ are defined by induction on $n$ as follows: $\overline{A}(b;)=\mathrm{Jb}$, and $\overline{A}(b;a_1,\dots ,a_{n+1})={\int }^x\overline{A}(x;a_1,\dots ,a_n)\otimes P(b,x\diamond a_{n+1})$.

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 promonidal category”.

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, An embedding theorem for closed categories

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