category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
Generalizing the classical notion of monoid, one can define a monoid (or monoid object) in any monoidal category $(C,\otimes,I)$. Classical monoids are of course just monoids in Set with the cartesian product.
By the microcosm principle, in order to define monoid objects in $C$, $C$ itself must be a “categorified monoid” in some way. The natural requirement is that it be a monoidal category. In fact, it suffices if $C$ is a multicategory. Contrast this with a group object, which can only be defined in a cartesian monoidal category (or a cartesian multicategory).
Namely, a monoid in $C$ is an object $M$ equipped with a multiplication $\mu: M \otimes M \to M$ and a unit $\eta: I \to M$ satisfying the associative law:
and the left and right unit laws:
Here $\alpha$ is the associator in $C$, while $\lambda$ and $\rho$ are the left and right unitors.
The analogue of a monoid homomorphism, called a morphism of monoids, is a morphism, $\f: M \to M'$ between two monoid objects, satisfying the equations;
$f \circ \mu = \mu' \circ (f \otimes f)$
$f \circ \eta = \eta'$
corresponding to the commutative diagrams;
Just as the category of regular monoids is equivalent to the category of locally small (i.e. Set-enriched) categories with one object, the category of monoids in $C$ (with the obvious morphisms) is equivalent to the category of $C$-enriched categories with one object.
A monoid in a category of modules is an associative unital algebra. A monoid in a category of endofunctors where tensor product is defined by composition, is a monad.
Categorical properties of monoid objects in monoidal categories are spelled out in sections 1.2 and 1.3 of
A summary is in section 4.1 of
See also MO/180673.