# nLab strict initial object

Contents

### Context

category theory

#### Limits and colimits

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

## Properties

• The Cartesian product of any object $X$ with a strict initial object is isomorphic to the strict initial object, $X \times \varnothing \simeq \varnothing$, because the projection $pr_1 \colon \varnothing \times X \to \varnothing$ exists by definition of Cartesian products, whence (1) implies that it is an isomorphism

$\varnothing \times X \underoverset {\;\;\;\simeq\;\;\;} {pr_1} {\longrightarrow} \varnothing \,.$
• Strict initial objects may be understood as van Kampen colimits, see e.g. Sobocinski & Heindel (2011), Exp. 4.5 (i). Indeed, the van Kampen-property in this case requires that the slice category $C/\varnothing$ be equivalent to the terminal category $\mathbf{1}$.

## Examples

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

Discussion as van Kampen colimits:

Last revised on March 8, 2024 at 13:02:15. See the history of this page for a list of all contributions to it.