Construction 2.2.1.1.2 (01EC): Construction of the Initial Set

Preferences





Font:

PDF Style:

Here's a breakdown of the differences between each PDF style:

Style	Class	Font	Theorem Environments
Style 1	`book`	Alegreya Sans	`tcbthm`
Style 2	`book`	Alegreya Sans	`amsthm`
Style 3	`book`	Arno*	`amsthm`
Style 4	`book`	Computer Modern	`amsthm`

*To be replaced with Linus Romer's Elemaints when it is released.

Table of Contents
Part 1: Sets
Chapter 2: Constructions With Sets
Section 2.2: Colimits of Sets
Subsection 2.2.1: The Initial Set
Construction 2.2.1.1.2: Construction of the Initial Set (cite)

Construction 2.2.1.1.2 ▶ Construction of the Initial Set

The initial set is the pair $\webleft (\text{Ø},\webleft\{ \iota _{A}\webright\} _{A\in \text{Obj}\webleft (\mathsf{Sets}\webright )}\webright )$ consisting of:

The Colimit. The empty set $\text{Ø}$ of Definition 2.3.1.1.1.
The Cocone. The collection of maps
\[ \webleft\{ \iota _{A}\colon \text{Ø}\to A\webright\} _{A\in \text{Obj}\webleft (\mathsf{Sets}\webright )} \]

given by the inclusion maps from $\text{Ø}$ to $A$.

Proof of Construction 2.2.1.1.2.

We claim that $\text{Ø}$ is the initial object of $\mathsf{Sets}$. Indeed, suppose we have a diagram of the form

in $\mathsf{Sets}$. Then there exists a unique map $\phi \colon \text{Ø}\to A$ making the diagram

commute, namely the inclusion map $\iota _{A}$.

Noticed something off, or have any comments? Feel free to reach out!

You can also use the contact form below: