nLab
strict initial object

Contents

Context

Category theory

Limits and colimits

Contents

Idea

The empty set, among all sets, has two characteristic properties:

  1. there is a unique function out of the empty set, to any other set;

  2. there is no function to the empty set, except from itself.

The first property generalizes to arbitrary categories as the property of an initial object.

The corresponding generalization including also the second property is that of a strict initial object:

Definition

An initial object \varnothing is called a strict initial object if every morphism to \varnothing is an isomorphism:

(1)XfXf. X \overset{f}{\longrightarrow} \varnothing \;\;\;\;\;\;\;\;\; \Rightarrow \;\;\;\;\;\;\;\;\; X \underoverset{\simeq}{f}{\longrightarrow} \emptyset \,.

Properties

  • The Cartesian product of any object XX with a strict initial object is isomorphic to the strict initial object, X×X \times \varnothing \simeq \varnothing, because the projection pr 1:×Xpr_1 \colon \varnothing \times X \to \varnothing exists by definition of Cartesian products, whence (1) implies that it is an isomorphism

    ×Xpr 1. \varnothing \times X \underoverset {\;\;\;\simeq\;\;\;} {pr_1} {\longrightarrow} \varnothing \,.

Examples

The empty set is a strict initial object in Sets.

The empty topological space is a strict initial object in TopologicalSpaces.

The empty groupoid is a strict initial object in Groupoids.

The empty simplicial set is a strict initial object in SimplicialSets.

The initial objects of any of the following types of categories are strict:

Specifically the initial objects of Set, Cat, Top are all strict.

At the other extreme, a zero object is only a strict initial object if the category is trivial (equivalent to the terminal category).

At the other extreme, a zero object is a strict initial object only if the category is trivial (i.e. equivalent to the terminal category).

References

Last revised on November 7, 2021 at 12:26:26. See the history of this page for a list of all contributions to it.