nLab
relevance monoidal category

Relevance monoidal categories

Context

Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

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, and seems to have been introduced in (Petrić 2002).

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

  • The category of pointed sets, which is equivalent to the category of sets and partial functions, is a relevance monoidal category with its pointed smash product (Došen-Petrić 2007). The unit for this tensor product is not the terminal object, which is the 1-element pointed set, but instead the 2-element pointed set, so this relevance monoidal category is not cartesian. The category of pointed sets can also be given a cartesian monoidal structure.

  • Any Church monoid is a relevance monoidal category whose underlying category is a poset.

References

  • B. Jacobs, Semantics of Weakening and Contraction, Annals of Pure and Applied Logic, volume 69, Issue 1, (1994) pp.73-106, https://doi.org/10.1016/0168-0072(94)90020-5

  • Z. Petrić, Coherence in Substructural Categories, Studia Logica volume 70, (2002) pp.271–296, doi:10.1023/A:1015186718090, arXiv:math/0006061

  • K. Došen and Z. Petrić, Relevant Categories and Partial Functions, Publications de l’Institut Mathématique, Nouvelle Série, Vol. 82(96), pp. 17–23 (2007) (doi:10.2298/PIM0796017D, arxiv:math/0504133)

Last revised on August 31, 2021 at 14:51:01. See the history of this page for a list of all contributions to it.