nLab
compact closed category

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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.

In particular, a compact closed category is a closed monoidal category, with the internal hom given by [A,B]=A *⊗B (where A * is the dual of A).

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.

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 June 18, 2012 08:42:17 by Cameron Smith (204.69.190.254)
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