category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
A compact closed category, or simply a compact category, is a symmetric monoidal category in which every object is dualizable, hence a rigid symmetric monoidal category.
More generally, if we drop the symmetry requirement, we obtain a rigid monoidal category, a.k.a. an autonomous category. Thus a compact category may also be called a rigid symmetric monoidal category or a symmetric autonomous category. A maximally clear, but rather verbose, term would be a symmetric monoidal category with duals for objects.
A rigid symmetric monoidal category $(\mathcal{C}, \otimes)$ is in particular a closed monoidal category, with the internal hom given by
(where $A^*$ is the dual object of $A$), via the adjunction natural equivalence that defines dual objects
This is what the terminology โcompact closedโ refers to.
The inclusion from the category of compact closed categories into the category of closed symmetric monoidal categories also has a left adjoint (Day 1977). Given a closed symmetric monoidal category $\mathcal{S}$, the free compact closed category $C(\mathcal{S})$ over $\mathcal{S}$ may be described as a localization of $\mathcal{S}$ by the maps
corresponding to the tensorial strength of the functors $[A,-] : \mathcal{S} \to \mathcal{S}$.
Given a traced monoidal category $\mathcal{C}$, there is a free construction completion of it to a compact closed category $Int(\mathcal{C})$ (Joyal-Street-Verity 96):
the objects of $Int(\mathcal{C})$ are pairs $(A^+, A^-)$ of objects of $\mathcal{C}$, a morphism $(A^+ , A^-) \to (B^+ , B^-)$ in $Int(\mathcal{C})$ is given by a morphism of the form $A^+\otimes B^- \longrightarrow A^- \otimes B^+$ in $\mathcal{C}$, and composition of two such morphisms $(A^+ , A^-) \to (B^+ , B^-)$ and $(B^+ , B^-) \to (C^+ , C^-)$ is given by tracing out $B^+$ and $B^-$ in the evident way.
A compact closed category is a star-autonomous category: the tensor unit is a dualizing object.
The category FinVect of finite-dimensional vector spaces is compact closed, see here.
A compact closed discrete category is just an abelian group.
The delooping $\mathbf{B}M$ of a commutative monoid $M$ is a compact closed category, and conversely, any compact closed category (or more generally, any closed monoidal category) with a single object must be isomorphic to the delooping of some commutative monoid.
The characterization of the free compact closed category over a closed symmetric monoidal category is described in
Discussion of coherence in compact closed categories is due to
The relation to quantum operations and completely positive maps is discussed in
The relation to traced monoidal categories is discussed in