Traced monoidal category
In category theory, a traced monoidal category is a category with some extra structure which gives a reasonable notion of feedback.
A traced symmetric monoidal category is a symmetric monoidal category C together with a family of functions
called a trace, satisfying the following conditions:
- naturality in : for every and ,

Naturality in X
- naturality in : for every and ,

Naturality in Y
- dinaturality in : for every and

Dinaturality in U
- vanishing I: for every , (with being the right unitor),

Vanishing I
- vanishing II: for every

Vanishing II
- superposing: for every and ,

Superposing
- yanking:
(where is the symmetry of the monoidal category).

Yanking
Properties
- Every compact closed category admits a trace.
- Given a traced monoidal category C, the Int construction generates the free (in some bicategorical sense) compact closure Int(C) of C.
References
- André Joyal, Ross Street, Dominic Verity (1996). "Traced monoidal categories". Mathematical Proceedings of the Cambridge Philosophical Society. 3: 447–468. doi:10.1017/S0305004100074338.CS1 maint: multiple names: authors list (link)
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