2.2.1 The Initial Set

Definition 2.2.1.1.1 ▶ The Initial Set

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

  • The Limit. The empty set $\emptyset $ of Definition 2.3.1.1.1.
  • The Cone. The collection of maps

    \[ \webleft\{ \iota _{A}\colon \emptyset \to A\webright\} _{A\in \text{Obj}\webleft (\mathsf{Sets}\webright )} \]

    given by the inclusion maps from $\emptyset $ to $A$.

Proof of Definition 2.2.1.1.1.

We claim that $\emptyset $ 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 $\phi \colon \emptyset \to A$ making the diagram

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

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