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

called a trace, satisfying the following conditions:

• naturality in : for every and ,

:

• naturality in : for every and ,

:

• dinaturality in : for every and

:

• vanishing I: for every , (with being the right unitor),

:

• vanishing II: for every

:

• superposing: for every and ,

:

• yanking:

:

(where 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