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 $$g:X' o X,::$$mathrm{Tr}^U_{X,Y}(f)g=mathrm{Tr}^U_{X',Y}(f(gotimes U))
* naturality in "Y": for every $$f:Xotimes U o Yotimes U and $$g:Y o Y',::$$gmathrm{Tr}^U_{X,Y}(f)=mathrm{Tr}^U_{X,Y'}((gotimes U)f)
* dinaturality in "U": for every $$f:Xotimes U o Yotimes U' and $$g:U' o U::$$mathrm{Tr}^U_{X,Y}((Yotimes g)f)=mathrm{Tr}^{U'}_{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