Item 1 (01QN) — The Clowder Project

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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.

Active-Inert Factorisation. Every morphism of pointed sets $f\colon \webleft (X,x_{0}\webright )\to \webleft (Y,y_{0}\webright )$ factors uniquely 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 an orthogonal factorisation system in $\mathsf{Sets}_{*}$.

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