4.1.5 Active and Inert Morphisms of Pointed Sets

Definition 4.1.5.1.1 ▶ Active and Inert Morphisms of Pointed Sets

Let $f\colon \webleft (X,x_{0}\webright )\to \webleft (Y,y_{0}\webright )$ be a morphism of pointed sets.

  1. The morphism $f$ is active if $f^{-1}\webleft (y_{0}\webright )=x_{0}$.
  2. The morphism $f$ is inert if, for each $y\in Y$, the set $f^{-1}\webleft (y\webright )$ has exactly one element.

Notation 4.1.5.1.2 ▶ The Category of Pointed Sets and Active Morphisms

We write $\mathsf{Sets}^{\mathrm{actv}}_{*}$ for the wide subcategory of $\mathsf{Sets}_{*}$ spanned by pointed sets and the active maps between them.

Example 4.1.5.1.3 ▶ Examples of Active and Inert Maps of Pointed Sets

Here are some examples of active and inert maps of pointed sets.

  1. The map $\mu \colon \left\langle 2\right\rangle \to \left\langle 1\right\rangle $ given by

    is active but not inert.

  2. The map $f\colon \left\langle 2\right\rangle \to \left\langle 2\right\rangle $ given by

    is inert but not active.

  3. The map $f\colon \left\langle 3\right\rangle \to \left\langle 1\right\rangle $ given by

    is neither inert nor active. However, it factors as $f=a\circ i$, where

    \begin{align*} i & \colon \left\langle 3\right\rangle \to \left\langle 2\right\rangle ,\\ a & \colon \left\langle 2\right\rangle \to \left\langle 1\right\rangle \end{align*}

    are the morphisms of pointed sets given by

    with $i$ being inert and $a$ being active.

Proposition 4.1.5.1.4 ▶ Properties of Active and Inert Maps of Pointed Sets

Let $\webleft (X,x_{0}\webright )$ and $\webleft (Y,y_{0}\webright )$ be pointed sets.

  1. Active-Inert Factorisation. Every morphism of pointed sets $f\colon \webleft (X,x_{0}\webright )\to \webleft (Y,y_{0}\webright )$ factors as
    \[ f=a\circ i, \]

    where:

    1. The map $i\colon \webleft (X,x_{0}\webright )\to \webleft (K,k_{0}\webright )$ is an inert morphism of pointed sets
    2. The map $a\colon \webleft (K,k_{0}\webright )\to \webleft (Y,y_{0}\webright )$ is an active morphism of pointed sets.

    Moreover, this determines a factorisation system in $\mathsf{Sets}_{*}$.

Proof of Proposition 4.1.5.1.4.

Item 1: Active-Inert Factorisation
Omitted.

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