A promonoidal category is like a monoidal category in whose structure (namely, tensor product and unit object) we have replaced functors by profunctors. It is a categorification of the idea of a boolean algebra.
Recalling that a profunctor is defined to be a functor , we can make this more explicit. We can also generalize it by replacing by a Benabou cosmos and by a -enriched category; then a profunctor is a -functor .
Thus, we obtain the following as an explicit definition of promonoidal -category: we have the following data
A -category .
A -ary functor . For notational clarity, we may write as .
A -functor .
and natural isomorphisms
satisfying the pentagon and unit axioms.
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 and are representable.
A promonoidal structure on suffices to induce a monoidal structure on by Day convolution. In fact, given a small -category , there is an equivalence of categories between
the category of pro-monoidal structures on , with strong pro-monoidal functors between them, and
the category of biclosed monoidal structures on , with strong monoidal functors between them.
A promonoidal structure on can be identified with a particular sort of multicategory structure on , i.e. with a co-multicategory structure on . The set is regarded as the set of co-multimorphisms .
More generally, we define a co-multicategory as follows. The objects of are the objects of . The co-multimorphisms in are defined by induction on as follows: , and .
Not every co-multicategory arises from a promonoidal one in this way. Roughly, a promonoidal category is a co-multicategory whose -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”.
Brian Day, An embedding theorem for closed categories
Day, Panchadcharam and Street, On centres and lax centres for promonoidal categories.