Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Exercise 9.6.0.5. Let $\operatorname{Set}$ denote the category of sets, let $\mathrm{Surj}$ be the collection of surjective functions, and let $\mathrm{Inj}$ be the collection of injective functions. Show that:

  • The pair $(\mathrm{Surj}, \mathrm{Inj} )$ is a factorization system on $\operatorname{Set}$.

  • The pair $( \mathrm{Inj}, \mathrm{Surj})$ is a weak factorization system on $\operatorname{Set}$ which is not a factorization system.

See Exercise 9.6.6.22.