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Traced monoidal category

From Wikipedia, the free encyclopedia

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 X
  • naturality in : for every and ,
Naturality in Y
Naturality in Y
  • dinaturality in : for every and
Dinaturality in U
Dinaturality in U
  • vanishing I: for every , (with being the right unitor),
Vanishing I
Vanishing I
  • vanishing II: for every
Vanishing II
Vanishing II
  • superposing: for every and ,
Superposing
Superposing
  • yanking:

(where is the symmetry of the monoidal category).

Yanking
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)
This page was last edited on 24 March 2020, at 22:24
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