category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
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).
Any Church monoid is a relevance monoidal category whose underlying category is a poset.
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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