Traced monoidal category

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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 ,
  • yanking:

(where is the symmetry of the monoidal category).



  • 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.


  • 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.