Definitions

# 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

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

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

$mathrm\left\{Tr\right\}^U_\left\{X,Y\right\}\left(f\right)g=mathrm\left\{Tr\right\}^U_\left\{X\text{'},Y\right\}\left(f\left(gotimes U\right)\right)$

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

$gmathrm\left\{Tr\right\}^U_\left\{X,Y\right\}\left(f\right)=mathrm\left\{Tr\right\}^U_\left\{X,Y\text{'}\right\}\left(\left(gotimes U\right)f\right)$

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

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

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

$mathrm\left\{Tr\right\}^I_\left\{X,Y\right\}\left(f\right)=f$

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

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

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

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

• yanking:

$mathrm\left\{Tr\right\}^U_\left\{U,U\right\}\left(gamma_\left\{U,U\right\}\right)=U$
(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

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