[FOM] Categories satisfying Schoeder-Bernstein theorem

joeshipman@aol.com joeshipman at aol.com
Wed May 13 00:03:47 EDT 2009

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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