nLab empty function

Definition

Given a set $X$, the empty function to $X$ is a function

$$\varnothing \longrightarrow X$$

to$X$ from the empty set.

This always exists and is unique; in other words, the empty set is an initial object in the category of sets.

If regarded as a bundle, the empty function is the empty bundle over its codomain.

In generalization to ambient categories other that Sets, an empty morphism would be any morphism out of a strict initial object.

Properties

The empty function to the empty set is not a constant function, though it is a weakly constant function.

Related concepts

• 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