The notion of $*$-category combines that of a $\dagger$-category and of an $\mathbb{C}$-algebroid (i.e. a $\mathbb{C}$-linear category). Thus it might reasonably be called simply a $\mathbb{C}$-linear $\dagger$-category. However, $*$-categories were popular in AQFT for a while, while the notion of dagger-category was introduced in the context of quantum information theory, so for historical reasons we have two names.

Definition

Definition

A $*$-category is a category that is a $\dagger$-category and a $\mathbb{C}$-algebroid such that the dagger operation is conjugate linear.

In a $*$-category, the dagger operation $\dagger$ is synonymously denoted $*$.