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.
In particular, a compact closed category is a closed monoidal category, with the internal hom given by (where is the dual object of ).
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.
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-autnomous categories
A compact closed category is a star-autonomous category with the tensor unit as dualizing object.
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