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 (where we sometimes denote an identity morphism by the corresponding object, e.g., using U to denote ${\text{id}}_{U}$):

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

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

Naturality in X

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

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

Naturality in Y

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

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

Dinaturality in U

• vanishing I: for every $f:X\otimes I\to Y\otimes I$,

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

Vanishing I

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

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

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:

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

(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

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