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

mathrm{Tr}^U_{X,Y}:mathbf{C}(Xotimes U,Yotimes U)tomathbf{C}(X,Y)
called a trace, satisfying the following conditions:

  • naturality in X: for every f:Xotimes Uto Yotimes U and g:X'to X,

mathrm{Tr}^U_{X,Y}(f)g=mathrm{Tr}^U_{X',Y}(f(gotimes U))

  • naturality in Y: for every f:Xotimes Uto Yotimes U and g:Yto Y',

gmathrm{Tr}^U_{X,Y}(f)=mathrm{Tr}^U_{X,Y'}((gotimes U)f)

  • dinaturality in U: for every f:Xotimes Uto Yotimes U' and g:U'to U

mathrm{Tr}^U_{X,Y}((Yotimes g)f)=mathrm{Tr}^{U'}_{X,Y}(f(Xotimes g))

  • vanishing I: for every f:Xotimes Ito Yotimes I,


  • vanishing II: for every f:Xotimes Uotimes Vto Yotimes Uotimes V

mathrm{Tr}^{Uotimes V}_{X,Y}(f)=mathrm{Tr}^U_{X,Y}(mathrm{Tr}^V_{X,Y}(f))

  • superposing: for every f:Xotimes Uto Yotimes U and g:Wto Z,

gotimes mathrm{Tr}^U_{X,Y}(f)=mathrm{Tr}^U_{Wotimes X,Zotimes Y}(gotimes f)

  • yanking:

(where gamma 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.


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