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:


 * naturality in $$X$$: for every $$f:X\otimes U\to Y\otimes U$$ and $$g:X'\to X$$,
 * $$\mathrm{Tr}^U_{X',Y}(f \circ (g\otimes \mathrm{id}_U)) = \mathrm{Tr}^U_{X,Y}(f) \circ g$$




 * naturality in $$Y$$: for every $$f:X\otimes U\to Y\otimes U$$ and $$g:Y\to Y'$$,
 * $$\mathrm{Tr}^U_{X,Y'}((g\otimes \mathrm{id}_U) \circ f) = g \circ \mathrm{Tr}^U_{X,Y}(f)$$




 * dinaturality in $$U$$: for every $$f:X\otimes U\to Y\otimes U'$$ and $$g:U'\to U$$
 * $$\mathrm{Tr}^U_{X,Y}((\mathrm{id}_Y\otimes g) \circ f)=\mathrm{Tr}^{U'}_{X,Y}(f \circ (\mathrm{id}_X\otimes g))$$




 * 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),
 * $$\mathrm{Tr}^I_{X,Y}(f)=\rho_Y \circ f \circ \rho_X^{-1}$$




 * vanishing II: for every $$f:X\otimes U\otimes V\to Y\otimes U\otimes V$$
 * $$\mathrm{Tr}^U_{X,Y}(\mathrm{Tr}^V_{X\otimes U,Y\otimes U}(f)) = \mathrm{Tr}^{U\otimes V}_{X,Y}(f)$$




 * superposing: for every $$f:X\otimes U\to Y\otimes U$$ and $$g:W\to Z$$,
 * $$g\otimes \mathrm{Tr}^U_{X,Y}(f)=\mathrm{Tr}^U_{W\otimes X,Z\otimes Y}(g\otimes f)$$




 * 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.