compact closed category


Monoidal categories



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.


Internal hom and compact closure

A rigid symmetric monoidal category (๐’ž,โŠ—)(\mathcal{C}, \otimes) is in particular a closed monoidal category, with the internal hom given by

[A,B]โ‰ƒA *โŠ—B [A,B] \simeq A^* \otimes B

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

๐’ž(C,[A,B])โ‰ƒ๐’ž(C,A *โŠ—B)โ‰ƒ๐’ž(CโŠ—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 โ€œcompact closedโ€ refers to.

The inclusion from the category of compact closed categories into the category of closed symmetric monoidal categories also has a left adjoint (Day 1977). Given a closed symmetric monoidal category ๐’ฎ\mathcal{S}, the free compact closed category C(๐’ฎ)C(\mathcal{S}) over ๐’ฎ\mathcal{S} may be described as a localization of ๐’ฎ\mathcal{S} by the maps

ฯƒ:[A,B]โŠ—Cโ†’[A,BโŠ—C] \sigma : [A,B] \otimes C \to [A, B \otimes C]

corresponding to the tensorial strength of the functors [A,โˆ’]:๐’ฎโ†’๐’ฎ[A,-] : \mathcal{S} \to \mathcal{S}.

Relation to traced monoidal categories

Given a traced monoidal category ๐’ž\mathcal{C}, there is a free construction completion of it to a compact closed category Int(๐’ž)Int(\mathcal{C}) (Joyal-Street-Verity 96):

the objects of Int(๐’ž)Int(\mathcal{C}) are pairs (A +,A โˆ’)(A^+, A^-) of objects of ๐’ž\mathcal{C}, a morphism (A +,A โˆ’)โ†’(B +,B โˆ’)(A^+ , A^-) \to (B^+ , B^-) in Int(๐’ž)Int(\mathcal{C}) is given by a morphism of the form A +โŠ—B โˆ’โŸถA โˆ’โŠ—B +A^+\otimes B^- \longrightarrow A^- \otimes B^+ in ๐’ž\mathcal{C}, and composition of two such morphisms (A +,A โˆ’)โ†’(B +,B โˆ’)(A^+ , A^-) \to (B^+ , B^-) and (B +,B โˆ’)โ†’(C +,C โˆ’)(B^+ , B^-) \to (C^+ , C^-) is given by tracing out B +B^+ and B โˆ’B^- in the evident way.

Relation to star-autonomous categories

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



The characterization of the free compact closed category over a closed symmetric monoidal category is described in

  • Brian Day, Note on compact closed categories, J. Austral. Math. Soc. 24 (Series A), 309-311 (1977)

Discussion of coherence in compact closed categories is due to

  • Max Kelly, M.L. Laplaza, Coherence for compact closed categories, Journal of Pure and Applied Algebra 19: 193โ€“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 February 11, 2016 06:15:19 by Noam Zeilberger (