Traced monoidal category

Last updated March 25, 2020

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 $Y$ : 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)

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))

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 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)

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)

yanking:

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

(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.

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References

André Joyal, Ross Street, Dominic Verity (1996). "Traced monoidal categories". Mathematical Proceedings of the Cambridge Philosophical Society. 3: 447–468. doi:10.1017/S0305004100074338.CS1 maint: multiple names: authors list (link)

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