Here are some examples of active and inert maps of pointed sets.
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The map $\mu \colon \left\langle 2\right\rangle \to \left\langle 1\right\rangle $ given by
is active but not inert.
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The map $f\colon \left\langle 2\right\rangle \to \left\langle 2\right\rangle $ given by
is inert but not active.
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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.
