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

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

called a trace, satisfying the following conditions:

• naturality in $X$: for every $f:X\otimes U\to Y\otimes U$ and $g:X'\to X$,

$\mathrm {Tr} _{X',Y}^{U}(f\circ (g\otimes \mathrm {id} _{U}))=\mathrm {Tr} _{X,Y}^{U}(f)\circ g$

Naturality in X

• naturality in $Y$: for every $f:X\otimes U\to Y\otimes U$ and $g:Y\to Y'$,

$\mathrm {Tr} _{X,Y'}^{U}((g\otimes \mathrm {id} _{U})\circ f)=g\circ \mathrm {Tr} _{X,Y}^{U}(f)$

Naturality in Y

• dinaturality in $U$: for every $f:X\otimes U\to Y\otimes U'$ and $g:U'\to U$

$\mathrm {Tr} _{X,Y}^{U}((\mathrm {id} _{Y}\otimes g)\circ f)=\mathrm {Tr} _{X,Y}^{U'}(f\circ (\mathrm {id} _{X}\otimes g))$

Dinaturality in U

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

$\mathrm {Tr} _{X,Y}^{I}(f)=\rho _{Y}\circ f\circ \rho _{X}^{-1}$

Vanishing I

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

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

Vanishing II

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

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

Superposing

• yanking:

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

(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

• Joyal, André; Street, Ross; Verity, Dominic (1996). "Traced monoidal categories". Mathematical Proceedings of the Cambridge Philosophical Society. 119 (3): 447–468. Bibcode:1996MPCPS.119..447J. doi:10.1017/S0305004100074338. S2CID 50511333.