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

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

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 $g:X'\to X$,

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

Naturality in X

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

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

Naturality in Y

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

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

Dinaturality in U

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

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

Vanishing I

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

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

Vanishing II

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

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

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.