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”.

References

  • Brian Day, An embedding theorem for closed categories

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

Revised on May 20, 2013 07:45:49 by Anonymous Coward (67.161.215.74)