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 relevance monoidal category is a symmetric monoidal category which has diagonal maps $A\to A\otimes A$, but not projection maps $A \to I$. (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).
Given a symmetric monoidal category $C$, let $CCSG(C)$ denote the category of commutative co-semigroups in $C$, i.e. objects $A$ equipped with a “comultiplication” $A\to A\otimes A$ that is coassociative and cocommutative. There is an obvious forgetful functor $CCSG(C) \to C$. Moreover, $CCSG(C)$ has a symmetric monoidal structure making the forgetful functor strict symmetric monoidal: if $A$ and $B$ are cosemigroups then $A\otimes B \to (A\otimes A)\otimes (B\otimes B) \cong (A\otimes B) \otimes (A\otimes B)$ makes $A\otimes B$ into a commutative cosemigroup as well, and the unit is the unit object $I$ of $C$ with its a canonical commutative cosemigroup structure given by the coherence isomorphism $I\cong I\otimes I$.
We say that $C$ is a relevance monoidal category if this functor $CCSG(C) \to C$ 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 $C$ a commutative cosemigroup structure on that object, in such a way that every morphism becomes a cosemigroup map, the structure on $A\otimes B$ is induced from those on $A$ and $B$ as above, and the structure on $I$ is the canonical one. This amounts to a natural assignment of “diagonal maps” $A\to A\otimes A$ 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 $C/I$ — 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.
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.
B. Jacobs, Semantics of Weakening and Contraction, Annals of Pure and Applied Logic, volume 69, Issue 1, (1994) pp.73-106, doi: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)
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