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 ,
![](https://upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Trace_diagram_naturality_1.svg/400px-Trace_diagram_naturality_1.svg.png)
- naturality in : for every and ,
![](https://upload.wikimedia.org/wikipedia/commons/thumb/2/26/Trace_diagram_naturality_2.svg/400px-Trace_diagram_naturality_2.svg.png)
- dinaturality in : for every and
![](https://upload.wikimedia.org/wikipedia/commons/thumb/8/84/Trace_diagram_dinaturality.svg/400px-Trace_diagram_dinaturality.svg.png)
- vanishing I: for every , (with being the right unitor),
![](https://upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Trace_diagram_vanishing.svg/400px-Trace_diagram_vanishing.svg.png)
- vanishing II: for every
![](https://upload.wikimedia.org/wikipedia/commons/thumb/8/81/Trace_diagram_associativity.svg/400px-Trace_diagram_associativity.svg.png)
- superposing: for every and ,
![](https://upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Trace_diagram_superposition.svg/400px-Trace_diagram_superposition.svg.png)
- yanking:
(where is the symmetry of the monoidal category).
![](https://upload.wikimedia.org/wikipedia/commons/thumb/5/52/Trace_diagram_yanking.svg/400px-Trace_diagram_yanking.svg.png)
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
- Joyal, André; Street, Ross; Verity, Dominic (1996). "Traced monoidal categories". Mathematical Proceedings of the Cambridge Philosophical Society. 119 (3): 447–468. Bibcode:1996MPCPS.119..447J. doi:10.1017/S0305004100074338. S2CID 50511333.