category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
The concept of traced monoidal category axiomatizes the structure on a monoidal category for it to have a sensible notion of trace the way it exists canonically in compact closed categories.
The original definition due to (Joyal-Street-Verity 96) is stated in the general setting of balanced monoidal categories. Here we give just the slightly simpler formulation for the case of symmetric monoidal categories (Hasegawa 1997).
A symmetric monoidal category $(C,\otimes,1,b)$ (where $b$ is the symmetry) is said to be traced if it is equipped with a natural family of functions
satisfying three axioms:
Vanishing: $Tr_{A,B}^1(f) = f$ (for all $f : A \to B$) and $Tr_{A,B}^{X\otimes Y}(f) = Tr_{A,B}^X(Tr_{A\otimes X,B\otimes X}^Y(f))$ (for all $f : A \otimes X \otimes Y \to B \otimes X \otimes Y$)
Superposing: $Tr_{C\otimes A,C\otimes B}^X(id_C \otimes f) = id_C \otimes Tr_{A,B}^X(f)$ (for all $f : A \otimes X \to B \otimes X$)
Yanking: $Tr_{X,X}^X(b_{X,X}) = id_X$
In string diagrams, the trace $Tr(f) : A \to B$ of a morphism $f : A \otimes X \to B \otimes X$ is visualized by wrapping the outgoing wire representing $X$ to the incoming wire representing $X$, thus “tying a loop” in the diagram of $f$. The three axioms above (as well as the naturality conditions) then all have natural graphical interpretations (see Joyal-Street-Verity 96 or Hasegawa 1997).
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.
For a cartesian monoidal category, the existence of a trace operator is equivalent to the existence of a “parameterized” fixed point operator satisfying certain properties (Hasegawa 1997).
Traced monoidal categories serve as an “operational” categorical semantics for linear logic, known as Geometry of Interactions. See there for more.
In this context the free compact closure $Int(\mathcal{C})$ from above is sometimes called the Geometry of Interaction construction and denoted $\mathcal{G}(\mathcal{C})$ (Abramsky-Haghverdi-Scott 02, def. 2.6).
The concept was introduced in
A characterization of trace structures on cartesian monoidal categories is given in
The equivalence between traces and parameterized fixed point operators appears as Theorem 3.1 (the author notes that this theorem was also proved independently by Martin Hyland).
Comprehensive discussion as a source for categorical semantics of the Geometry of Interactions is in