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:$mathrm\{Tr\}^U\_\{X,Y\}:mathbf\{C\}(Xotimes\; U,Yotimes\; U)\; omathbf\{C\}(X,Y)$called a "trace", satisfying the following conditions:
* naturality in "X": for every $f:Xotimes\; U\; o\; Yotimes\; U$ and $g:X\text{'}\; o\; X$,::$mathrm\{Tr\}^U\_\{X,Y\}(f)g=mathrm\{Tr\}^U\_\{X\text{'},Y\}(f(gotimes\; U))$
* naturality in "Y": for every $f:Xotimes\; U\; o\; Yotimes\; U$ and $g:Y\; o\; Y\text{'}$,::$gmathrm\{Tr\}^U\_\{X,Y\}(f)=mathrm\{Tr\}^U\_\{X,Y\text{'}\}((gotimes\; U)f)$
* dinaturality in "U": for every $f:Xotimes\; U\; o\; Yotimes\; U\text{'}$ and $g:U\text{'}\; o\; U$::$mathrm\{Tr\}^U\_\{X,Y\}((Yotimes\; g)f)=mathrm\{Tr\}^\{U\text{'}\}\_\{X,Y\}(f(Xotimes\; g))$
* vanishing I: for every $f:Xotimes\; I\; o\; Yotimes\; I$,::$mathrm\{Tr\}^I\_\{X,Y\}(f)=f$
* vanishing II: for every $f:Xotimes\; Uotimes\; V\; o\; Yotimes\; Uotimes\; V$::$mathrm\{Tr\}^\{Uotimes\; V\}\_\{X,Y\}(f)=mathrm\{Tr\}^U\_\{X,Y\}(mathrm\{Tr\}^V\_\{X,Y\}(f))$
* superposing: for every $f:Xotimes\; U\; o\; Yotimes\; U$ and $g:W\; o\; Z$,::$gotimes\; mathrm\{Tr\}^U\_\{X,Y\}(f)=mathrm\{Tr\}^U\_\{Wotimes\; X,Zotimes\; Y\}(gotimes\; f)$
* 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

* cite journal
author = André Joyal, Ross Street, Dominic Verity
year = 1996
title = Traced monoidal categories
journal = Mathematical Proceedings of the Cambridge Philosophical Society
volume = 3
pages = 447–468