# nLab compact closed category

### Context

#### Monoidal categories

monoidal categories

## With traces

• trace

• traced monoidal category?

# Contents

## Definition

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 $\left[A,B\right]={A}^{*}\otimes B$ (where ${A}^{*}$ is the dual object 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

• Max Kelly, M.L. Laplaza, Coherence for compact closed categories, Journal of Pure and Applied Algebra 19: 193–213 (1980)

section 2.1 in

Revised on October 4, 2013 04:01:48 by Urs Schreiber (188.200.54.65)