# nLab compact closed category

### Context

#### Monoidal categories

monoidal categories

# 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 $[A,B] = A^* \otimes B$ (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.

## Properties

### Relation to traced monoidal categories

Given a traced monoidal category $\mathcal{C}$, there is a free construction completion of it to a compact closed category $Int(\mathcal{C})$ (Joyal-Street-Verity 96):

the objects of $Int(\mathcal{C})$ are pairs $(A^+, A^-)$ of objects of $\mathcal{C}$, a morphism $(A^+ , A^-) \to (B^+ , B^-)$ in $Int(\mathcal{C})$ is given by a morphism of the form $A^+\otimes B^- \longrightarrow A^- \otimes B^+$ in $\mathcal{C}$, and composition of two such morphisms $(A^+ , A^-) \to (B^+ , B^-)$ and $(B^+ , B^-) \to (C^+ , C^-)$ is given by tracing out $B^+$ and $B^-$ 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.

## References

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: 193β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

Revised on January 29, 2014 11:05:56 by Anonymous Coward (198.129.67.84)