Subsection 4.1.2 (009H): Morphisms of Pointed Sets

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Table of Contents
Part 1: Sets
Chapter 4: Pointed Sets
Section 4.1: Pointed Sets
Subsection 4.1.2: Morphisms of Pointed Sets (cite)

4.1.2 Morphisms of Pointed Sets

Definition 4.1.2.1.1 ▶ Morphisms of Pointed Sets

A morphism of pointed sets¹^,² is equivalently:

A morphism of $\mathbb {E}_{0}$-monoids in $\webleft (\mathrm{N}_{\bullet }\webleft (\mathsf{Sets}\webright ),\text{pt}\webright )$.
A morphism of pointed objects in $\webleft (\mathsf{Sets},\text{pt}\webright )$.

¹Further Terminology: Also called a pointed function.

²Further Terminology: In the context of monoids with zero as models for $\mathbb {F}_{1}$-algebras, morphisms of pointed sets are also called morphism of $\mathbb {F}_{1}$-modules.

Remark 4.1.2.1.2 ▶ Unwinding Definition 4.1.2.1.1

In detail, a morphism of pointed sets $f\colon \webleft (X,x_{0}\webright )\to \webleft (Y,y_{0}\webright )$ is a morphism of sets $f\colon X\to Y$ such that the diagram

commutes, i.e. such that

\[ f\webleft (x_{0}\webright ) = y_{0}. \]

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