nLab
promonoidal category

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. It is a categorification of the idea of a boolean algebra.

Definition

A promonoidal category is a pseudomonoid in the monoidal bicategory Prof. This means that it is a category AA together with

  • A profunctor P:A×AAP \colon A\times A ⇸ A.
  • A profunctor J:1AJ\colon 1 ⇸ A.
  • Associativity and unit isomorphisms P(P×1)P(1×P)P \odot (P\times 1) \cong P\odot (1\times P), P(J×1)1P\odot (J\times 1) \cong 1, and P(1×J)1P\odot (1\times J) \cong 1.
  • The usual pentagon and unit conditions hold, as in a monoidal category.

Recalling that a profunctor ABA ⇸ B is defined to be a functor B op×ASetB^{op}\times A \to Set, we can make this more explicit. We can also generalize it by replacing SetSet by a Benabou cosmos VV and AA by a VV-enriched category; then a profunctor is a VV-functor B op×AVB^{op}\times A \to V.

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

  1. A VV-category AA.

  2. A 33-ary functor P:A opAAVP:A^\op \otimes A \otimes A\to V. For notational clarity, we may write P(a,b,c)P(a,b,c) as P(a,bc)P(a,b \diamond c).

  3. A VV-functor J:A opVJ:A^{op}\to V.

and natural isomorphisms

  1. λ ab: x(J(x)P(b,ax))A(b,a)\lambda_{ab}:\int^x (J(x) \otimes P(b,a \diamond x))\to A(b,a)

  2. ρ ab: x(J(x)P(b,xa))A(b,a)\rho_{ab}: \int^x ( J(x)\otimes P(b,x \diamond a))\to A(b,a)

  3. α abcd: x(P(x,ab)P(d,xc)) x(P(x,bc)P(d,ax))\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.

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 PP and JJ are representable.

Day convolution

A promonoidal structure on AA suffices to induce a monoidal structure on V A opV^{A^{op}} by Day convolution. In fact, given a small VV-category AA, there is an equivalence of categories between

  1. the category of pro-monoidal structures on AA, with strong pro-monoidal functors between them, and

  2. the category of biclosed monoidal structures on V A opV^{A^{op}}, with strong monoidal functors between them.

Versus multicategories

A promonoidal structure on AA can be identified with a particular sort of multicategory structure on A opA^{op}, i.e. with a co-multicategory structure on AA. The set P(x,y,z)P(x, y, z) is regarded as the set of co-multimorphisms x(y,z)x \to (y,z).

More generally, we define a co-multicategory A¯\bar A as follows. The objects of A¯\bar A are the objects of AA. The co-multimorphisms ba 1a nb\to a_1\dots a_n in A¯\bar A are defined by induction on nn as follows: A¯(b;)=Jb\bar A(b;)=Jb, and A¯(b;a 1,,a n+1)= xA¯(x;a 1,,a n)P(b,xa n+1)\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 nn-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”.

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 λ,ρ,α\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 CC corresponds to a pseudomonoid structure on C opC^{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).

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.

Revised on April 13, 2015 19:32:37 by Noam Zeilberger (176.189.43.179)