nLab
compact closed category

Contents

Definition

A compact closed category, or simply a compact category, is a symmetric monoidal category in which every object is dualizable, hence a rigid symmetric monoidal category.

In particular, a compact closed category is a closed monoidal category, with the internal hom given by [A,B]=A *βŠ—B[A,B] = A^* \otimes B (where A *A^* is the dual object of AA).

More generally, if we drop the symmetry requirement, we obtain a rigid monoidal category, a.k.a. an autonomous category. Thus a compact category may also be called a rigid symmetric monoidal category or a symmetric autonomous category. A maximally clear, but rather verbose, term would be a symmetric monoidal category with duals for objects.

Properties

Relation to traced monoidal categories

Given a traced monoidal category π’ž\mathcal{C}, there is a free construction completion of it to a compact closed category Int(π’ž)Int(\mathcal{C}) (Joyal-Street-Verity 96):

the objects of Int(π’ž)Int(\mathcal{C}) are pairs (A +,A βˆ’)(A^+, A^-) of objects of π’ž\mathcal{C}, a morphism (A +,A βˆ’)β†’(B +,B βˆ’)(A^+ , A^-) \to (B^+ , B^-) in Int(π’ž)Int(\mathcal{C}) is given by a morphism of the form A +βŠ—B βˆ’βŸΆA βˆ’βŠ—B +A^+\otimes B^- \longrightarrow A^- \otimes B^+ in π’ž\mathcal{C}, and composition of two such morphisms (A +,A βˆ’)β†’(B +,B βˆ’)(A^+ , A^-) \to (B^+ , B^-) and (B +,B βˆ’)β†’(C +,C βˆ’)(B^+ , B^-) \to (C^+ , C^-) is given by tracing out B +B^+ and B βˆ’B^- in the evident way.

Relation to star-autnomous categories

A compact closed category is a star-autonomous category: the tensor unit is a dualizing object.

Examples

References

Discussion of coherence in compact closed catories is due to

  • Max Kelly, M.L. Laplaza, Coherence for compact closed categories, Journal of Pure and Applied Algebra 19: 193–213 (1980)

The relation to quantum operations and completely positive maps is discussed in

  • Peter Selinger, Dagger compact closed categories and completely positive maps. pdf

The relation to traced monoidal categories is discussed in

Revised on July 31, 2014 02:39:59 by Sam? (131.174.142.158)