Concept

Traced monoidal category

Summary
In , 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 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)
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