category theory

# Contents

## Definition

###### Definition

A $C^*$-category is a $*$-category that has a norm on each set of arrows $Hom(x, y)$ that turns it into a Banach space with ${\|s^* s\|} = {\|s\|}^2$ for $s \in Hom(x, y)$ and ${\|{s t}\|} \leq {\|s\|} {\|t\|}$ for all arrows $s, t$ that are composable, that is $s \in Hom(x, y)$ and $t \in Hom(y, x)$.

###### Remark

A C-star-algebra $A$ is equivalently a pointed one-object $C^*$-category $\mathbf{B}A$ (the delooping of $A$). Accordingly, a more systematic name for $C^*$-categories is a $C^*$-algebroids.

## Examples

###### Example

The $C^\ast$-representation category of a weak Hopf C-star-algebra (see there for details) is naturally a rigid monoidal $C^\ast$-category.

Revised on April 5, 2013 18:58:44 by Ivanych? (31.162.96.145)