nLab
empty category

Contents

Definition

The empty category is the category with no objects, and hence no morphisms, i.e. the catgegory whose class of objects is the empty set.

This is the initial object in Cat.

This is a groupoid, so we may call it the empty groupoid. One may similarly speak of the empty n-category, the empty infinity-groupoid, etc etc etc.

Properties

The empty category is discrete, hence may be identified with a set: the empty set. This set is a subsingleton, so we may also identify it with a truth value: the false one.

The empty category is initial in Cat (as well as Grpd, ∞Grpd, Set, etc).

Last revised on June 12, 2018 at 12:11:59. See the history of this page for a list of all contributions to it.