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 $g:X\text{'}to\; X$,

$mathrm\{Tr\}^U\_\{X,Y\}(f)g=mathrm\{Tr\}^U\_\{X\text{'},Y\}(f(gotimes\; U))$

• naturality in Y: for every $f:Xotimes\; Uto\; Yotimes\; U$ and $g:Yto\; Y\text{'}$,

$gmathrm\{Tr\}^U\_\{X,Y\}(f)=mathrm\{Tr\}^U\_\{X,Y\text{'}\}((gotimes\; U)f)$

• dinaturality in U: for every $f:Xotimes\; Uto\; Yotimes\; U\text{'}$ and $g:U\text{'}to\; U$

$mathrm\{Tr\}^U\_\{X,Y\}((Yotimes\; g)f)=mathrm\{Tr\}^\{U\text{'}\}\_\{X,Y\}(f(Xotimes\; g))$

• vanishing I: for every $f:Xotimes\; Ito\; Yotimes\; I$,

$mathrm\{Tr\}^I\_\{X,Y\}(f)=f$

• 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:

$mathrm\{Tr\}^U\_\{U,U\}(gamma\_\{U,U\})=U$

(where $gamma$ is the symmetry of the monoidal category).

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.