With duals for objects
With duals for morphisms
Special sorts of products
In higher category theory
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 is in particular a closed monoidal category, with the internal hom given by
(where is the dual object of ), via the adjunction natural equivalence that defines dual objects
This is what the terminology “compact closed” 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 , the free compact closed category over may be described as a localization of by the maps
corresponding to the tensorial strength of the functors .
Relation to traced monoidal categories
Given a traced monoidal category , there is a free construction completion of it to a compact closed category (Joyal-Street-Verity 96):
the objects of are pairs of objects of , a morphism in is given by a morphism of the form in , and composition of two such morphisms and is given by tracing out and 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: 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