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If $\mathcal{C}$ is a locally presentable category, then it seems to be well-known that (Strong Epi, Mono) is a factorization system on $\mathcal{C}$. Where can I find a proof of this fact? Actually I only would like to see a proof that every morphism can be factored as an epimorphism followed by a monomorphism.

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up vote 2 down vote accepted

This is Proposition 1.61 in [Adámek and Rosický, Locally presentable and accessible categories]. The proof given merely observes that $\mathcal{C}$ is cocomplete, well copowered, and has pullbacks – so it has a (strong epi, mono) factorisation system. Dually, $\mathcal{C}$ has an (epi, strong mono) factorisation system, because it is complete, well powered, and has pushouts. This in turn is Proposition 4.4.3 in [Borceux, Handbook of categorical algebra, Vol. I].

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Thank you. $\phantom{ }$ – Martin Brandenburg Oct 27 '13 at 10:44

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