nLab empty category

Definition

The empty category is the category with no objects, and hence no morphisms, i.e. the category 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).

Related concepts

• terminal category

• empty set, empty subset

• empty type, empty context

• empty family, empty function, empty bundle

• empty topological space, empty simplicial set

• empty graph, empty category, empty diagram

• nothing