# [FOM] Categories satisfying Schoeder-Bernstein theorem

Jaap van Oosten J.vanOosten at uu.nl
Thu May 14 05:45:55 EDT 2009

```Dear Joe,

there is a categorical proof (in Boolean toposes) in Peter Johnstone's
Sketches of an Elephant, D4.1.11. He also shows that the assumption of
Booleanness is not a necessary condition, and gives a consequence of
Schroder-Bernstein in toposes with natural numbers objects.

Jaap van Oosten
joeshipman at aol.com wrote:
> What conditions must a category satisfy for the Schroder-Bernstein
> theorem to be true? (In categorial language, the Schroeder-Bernstein
> theorem holds if whenever there are monics f:A-->B and g:B-->A, there
> is an iso h:A-->B.) For the category of sets I know how to prove
> Schroder-Bernstein but I don't know a "categorial" proof.
>
> The dual version, which uses epics instead of monics, is obviously
> harder in the case of the category of sets because you can prove
> Schroeder-Bernstein without AC but you can't do it with surjections
> rather than injections unless you use AC. So something deep is going on
> here -- a categorial version would have to either use a categorial form
> of AC or else use assumptions that don't get preserved when the arrows
> are reversed.
>
> -- JS
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```