[Math] why split epi and mono implies iso


I was doing some exercises on the definitions of epics, monos, split monos, etc…, and I asked myself that if you could take, for instance an epi which is mono, and then deduce it is an iso, which is of course false; trying to recover the fact that in $Set$ epi and mono imply iso. So I wanna ask, if mono and coequalizer imply iso, or mono and split epi imply iso, or if both are false. I just wondered, since both coequalizer and split epi are stronger than epi; I've tried to show either, with no success.


Best Answer

If $f$ is a coequalizer of maps $g, h$ then $fg = fh$. If $f$ is also mono then $g = h$. The coequalizer of the pair $(g, g)$ is the identity map which is indeed an isomorphism. So mono + coequalizer does mean isomorphism.

For mono and split epi assume $fg = \mathrm{id}$. Then $fgf = f$ so if $f$ is mono $gf = \mathrm{id}$, hence $f$ is iso. Alternatively, observe that $f$ is a coequalizer for $(gf, \mathrm{id})$.

The same holds for epi + equalizer and epi + split mono.

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