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

\mathrm{Tr}^U_{X,Y}:\mathbf{C}(X\otimes U,Y\otimes U)\to\mathbf{C}(X,Y)

called a trace, satisfying the following conditions:

  • naturality in X: for every f:X\otimes U\to Y\otimes Y and g:X'\to X,
\mathrm{Tr}^U_{X,Y}(f)g=\mathrm{Tr}^U_{X',Y}(f(g\otimes U))
  • naturality in Y: for every f:X\otimes U\to Y\otimes U and g:Y\to Y',
g\mathrm{Tr}^U_{X,Y}(f)=\mathrm{Tr}^U_{X,Y'}((g\otimes U)f)
  • dinaturality in U: for every f:X\otimes U\to Y\otimes U' and g:U'\to U
\mathrm{Tr}^U_{X,Y}((Y\otimes g)f)=\mathrm{Tr}^{U'}_{X,Y}(f(X\otimes g))
  • vanishing I: for every f:X\otimes I\to Y\otimes I,
\mathrm{Tr}^I_{X,Y}(f)=f
  • vanishing II: for every f:X\otimes U\otimes V\to Y\otimes U\otimes V
\mathrm{Tr}^{U\otimes V}_{X,Y}(f)=\mathrm{Tr}^U_{X,Y}(\mathrm{Tr}^V_{X,Y}(f))
  • superposing: for every f:X\otimes U\to Y\otimes U and g:W\to Z,
g\otimes \mathrm{Tr}^U_{X,Y}(f)=\mathrm{Tr}^U_{W\otimes X,Z\otimes Y}(g\otimes f)
  • yanking:
\mathrm{Tr}^U_{U,U}(\gamma_{U,U})=U

(where γ is the symmetry of the monoidal category).

[edit] Properties

  • Given a traced monoidal category C, the Int construction generates the free (in some bicategorical sense) compact closure Int(C) of C.

[edit] References

  • André Joyal, Ross Street, Dominic Verity (1996). "Traced monoidal categories". Mathematical Proceedings of the Cambridge Philosophical Society 3: 447-468. 


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