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compact closed category

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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.

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

Internal hom and compact closure

A rigid symmetric monoidal category (π’ž,βŠ—)(\mathcal{C}, \otimes) is in particular a closed monoidal category, with the internal hom given by

[A,B]≃A *βŠ—B [A,B] \simeq A^* \otimes B

(where A *A^* is the dual object of AA), via the natural equivalence

π’ž(C,[A,B])β‰ƒπ’ž(C,A *βŠ—B)β‰ƒπ’ž(CβŠ—A,B). \mathcal{C}(C,[A,B]) \simeq \mathcal{C}(C, A^\ast \otimes B) \simeq \mathcal{C}(C \otimes A, B) \,.

This is what the terminology β€œcompact closed” refers to.

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 November 12, 2014 08:45:28 by Urs Schreiber (86.212.79.144)