nLab
relevance monoidal category

Relevance monoidal categories

Idea

A relevance monoidal category is a symmetric monoidal category which has diagonal maps AAAA\to A\otimes A, but not projection maps AIA\to I. (If it had both, it would be a cartesian monoidal category; while if it had projection maps but not diagonals it would be a semicartesian monoidal category.) The name comes from the connection with relevance logic.

Definition

Given a symmetric monoidal category CC, let CCSG(C)CCSG(C) denote the category of commutative co-semigroups in CC, i.e. objects AA equipped with a “comultiplication AAAA\to A\otimes A that is coassociative and cocommutative. There is an obvious forgetful functor CCSG(C)CCCSG(C) \to C. Moreover, CCSG(C)CCSG(C) has a symmetric monoidal structure making the forgetful functor strict symmetric monoidal: if AA and BB are cosemigroups then AB(AA)(BB)(AB)(AB)A\otimes B \to (A\otimes A)\otimes (B\otimes B) \cong (A\otimes B) \otimes (A\otimes B) makes ABA\otimes B into a commutative cosemigroup as well, and the unit is the unit object II of CC with its a canonical commutative cosemigroup structure given by the coherence isomorphism IIII\cong I\otimes I.

We say that CC is a relevance monoidal category if this functor CCSG(C)CCCSG(C) \to C is equipped with a strict section that is also a strict symmetric monoidal functor. That is, we have a functor assigning to every object of CC a commutative cosemigroup structure on that object, in such a way that every morphism becomes a cosemigroup map, the structure on ABA\otimes B is induced from those on AA and BB as above, and the structure on II is the canonical one. This amounts to a natural assignment of “diagonal maps” AAAA\to A\otimes A satisfying some straightforward axioms.

(If we replaced cosemigroups with comonoids, then the analogous property would characterize cartesian monoidal categories, while if instead we used copointed objects — that is, the slice C/IC/I — it would characterize semicartesian monoidal categories. Interestingly, unlike in those two cases, the relevance case doesn’t seem to imply any universal property for the monoidal product or the unit.)

One can of course additionally ask that a relevance monoidal category be closed, or that it have finite products or coproducts. One might also ask it to be star-autonomous, although in that case there might need to be some compatibility between the star-autonomy and the relevance.

Examples

References

  • K. Dosen, Z. Petric, Relevant Categories and Partial Functions, Publications de l’Institut Mathématique, Nouvelle Série, Vol. 82(96), pp. 17–23 (2007) arXiv

Revised on August 25, 2016 23:06:55 by Mike Shulman (76.167.222.204)