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

${\mathrm{T}\mathrm{r}}_{X,Y}^U:𝐂(X\otimes U,Y\otimes U)\to 𝐂(X,Y)$

called a trace, satisfying the following conditions:

• naturality in $X$: for every $f:X\otimes U\to Y\otimes U$ and $g:X^{\prime }\to X$,

${\mathrm{T}\mathrm{r}}_{X^{\prime },Y}^U(f\circ (g\otimes {\mathrm{i}\mathrm{d}}_U))={\mathrm{T}\mathrm{r}}_{X,Y}^U(f)\circ g$

Naturality in X

• naturality in $Y$: for every $f:X\otimes U\to Y\otimes U$ and $g:Y\to Y^{\prime }$,

${\mathrm{T}\mathrm{r}}_{X,Y^{\prime }}^U((g\otimes {\mathrm{i}\mathrm{d}}_U)\circ f)=g\circ {\mathrm{T}\mathrm{r}}_{X,Y}^U(f)$

Naturality in Y

• dinaturality in $U$: for every $f:X\otimes U\to Y\otimes U^{\prime }$ and $g:U^{\prime }\to U$

${\mathrm{T}\mathrm{r}}_{X,Y}^U(({\mathrm{i}\mathrm{d}}_Y\otimes g)\circ f)={\mathrm{T}\mathrm{r}}_{X,Y}^{U^{\prime }}(f\circ ({\mathrm{i}\mathrm{d}}_X\otimes g))$

Dinaturality in U

• vanishing I: for every $f:X\otimes I\to Y\otimes I$, (with ${\rho }_X:X\otimes I\cong X$ being the right unitor),

${\mathrm{T}\mathrm{r}}_{X,Y}^I(f)={\rho }_Y\circ f\circ {\rho }_X^{-1}$

Vanishing I

• vanishing II: for every $f:X\otimes U\otimes V\to Y\otimes U\otimes V$

${\mathrm{T}\mathrm{r}}_{X,Y}^U({\mathrm{T}\mathrm{r}}_{X\otimes U,Y\otimes U}^V(f))={\mathrm{T}\mathrm{r}}_{X,Y}^{U\otimes V}(f)$

Vanishing II

• superposing: for every $f:X\otimes U\to Y\otimes U$ and $g:W\to Z$,

$g\otimes {\mathrm{T}\mathrm{r}}_{X,Y}^U(f)={\mathrm{T}\mathrm{r}}_{W\otimes X,Z\otimes Y}^U(g\otimes f)$

Superposing

• yanking:

${\mathrm{T}\mathrm{r}}_{X,X}^X({\gamma }_{X,X})={\mathrm{i}\mathrm{d}}_X$

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

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

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