empty function in nLab

empty function in nLab

Contents

Context

Foundations

foundations

The basis of it all

Set theory

fundamentals of set theory
material set theory
presentations of set theory
structuralism in set theory
- material set theory
- structural set theory
  - categorical set theory
    - ETCS
    - structural ZFC
  - allegorical set theory
    - SEAR
class-set theory
constructive set theory
algebraic set theory

Foundational axioms

foundational axioms

Removing axioms

Contents

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 than 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.

Last revised on April 17, 2025 at 06:48:11. See the history of this page for a list of all contributions to it.