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 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 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 corresponds to a pseudomonoid structure on 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).
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.