With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
A promonoidal category is like a monoidal category in whose structure (namely, tensor product and unit object) we have replaced functors by profunctors.
A promonoidal category is a pseudomonoid in the monoidal bicategory Prof. This means that it is a category together with
Recalling that a profunctor ⇸ is defined to be a functor of the form , we can make this more explicit. We can also generalize it by replacing Set by a Benabou cosmos and by a -enriched category; then a profunctor is a -enriched functor .
Thus, we obtain the following as an explicit definition of promonoidal -category:
We have the following data
A -enriched category .
A -ary enriched functor . For notational clarity, we may write as .
A -functor .
and enriched natural isomorphisms
satisfying the pentagon and unit axioms for promonoidal categories. Explicitly, writting for , for , for , and for composition of profunctors, we require the following conditions to hold:
The triangle identity for promonoidal categories. For each , the diagram
The pentagon identity for promonoidal categories. For each , the diagram
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 promonoidal category”.
In fact, promonoidal categories correspond exactly to the exponentiable? multicategories: see multicategory for more information.
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).
Regarding monoidal categories as promonoidal is useful in order to express extra structure on them, such as closedness, -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.
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:
Last revised on September 7, 2024 at 10:29:24. See the history of this page for a list of all contributions to it.