In category theory

Category theory

Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

, 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 X: for every and ,

• naturality in Y: for every and ,

• dinaturality in U: for every and

• vanishing I: for every ,

• vanishing II: for every

• superposing: for every and ,

• yanking:

(where is the symmetry of the monoidal category).

Properties

• Every compact closed category
  Compact closed category
  In category theory, compact closed categories are a general context for treating dual objects. The idea of a dual object generalizes the more familiar concept of the dual of a finite-dimensional vector space...
  
  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.