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 (where we sometimes denote an identity morphism by the corresponding object, e.g., using U to denote ):

  • naturality in X: for every and ,
Naturality in X
  • naturality in Y: for every and ,
Naturality in Y
  • dinaturality in U: for every and
Dinaturality in U
  • vanishing I: for every ,
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.


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