nLab
compact closed category

Skip the Navigation Links | Home Page | All Pages | Recently Revised | Authors | Feeds | Export |

Context

Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

  • trace

  • traced monoidal category?

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Examples

Theorems

In higher category theory

Contents

Definition

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

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.

References

The Wikipedia entry isn’t bad:

Also see section 2.1 in:

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

Revised on December 16, 2010 14:37:49 by Urs Schreiber (131.211.233.8)
Edit | Back in time (8 revisions) | See changes | History | Views: Print | TeX | Source | Linked from: Timeline of category theory and related mathematics, Day convolution, monoidal category, tensor product, symmetric monoidal category, braided monoidal category, multicategory, periodic table, cartesian monoidal category, k-tuply monoidal n-category, cartesian closed category, locally cartesian closed category, Crans-Gray tensor product, red herring principle, monoidal model category, pushout-product axiom, closed monoidal category, strict monoidal category, dagger-compact category, span trace, hyperstructure, trace, dualizable object, string diagram, compact closed category, k-tuply monoidal (n,r)-category, star-autonomous category, closed monoidal structure on presheaves, exponential object, monoidal (infinity,1)-category, symmetric monoidal (infinity,1)-category, fusion category, rigid monoidal category, semisimple category, modular tensor category, symmetric monoidal functor, semicartesian monoidal category, 2009 September changes, cartesian bicategory, braided monoidal 2-category, trace of a category, monoidal functor, oplax monoidal functor, extranatural transformation, internal category in a monoidal category, coherence theorem for monoidal categories, tensor power, monoidal categories - contents, closed functor, monoidal bicategory, pivotal category, bicartesian closed category, quantum information, quantum mechanics in terms of dagger-compact categories, compact double category, symmetric monoidal dagger-category, symmetric algebra, ribbon category, category with duals, twist, bimonoidal category, spherical category, monoid axiom in a monoidal model category, symmetric monoidal (infinity,n)-category, model structure on monoids in a monoidal model category, super infinity-groupoid, premonoidal category, cartesian functor, monoidal structure map, cartesian closed (infinity,1)-category, category of monoids, closed monoidal (infinity,1)-category, braided monoidal functor, Hopf adjunction, evaluation map, Frobenius monoidal functor, bilax monoidal functor, monoidal Quillen adjunction, commutative monoid in a symmetric monoidal (infinity,1)-category, fully dualizable object, (infinity,n)-category with duals, Spanier-Whitehead duality, monoidal derivator, Picard group, contents of contents, Mac Lane's proof of the coherence theorem for monoidal categories, cartesian closed functor