[FOM] Fwd: Categories satisfying Schoeder-Bernstein theorem

[Thomas Forster]

Peter Johnstone has contributed this.  `The Elephant' is his - 
elephantine - book about Topoi.


On Thu, 14 May 2009, Prof. Peter Johnstone 
wrote:

> There's a relevant paper by Banaschewski & Brummer: "Thoughts on the
> Cantor--Bernstein Theorem", Quaest. Math. 9 (1986), 1--27. There are
> also some comments on pages 950--1 of the Elephant.
>
> Peter
>
> On Thu, 14 May 2009, T.Forster at dpmms.cam.ac.uk wrote:
>
>> would you care to comment?
>> 
>> 
>> ---------- Forwarded message ----------
>> To: FOM at cs.nyu.edu
>> Date: Wed, 13 May 2009 00:03:47 -0400
>> From: joeshipman at aol.com
>> Subject: [FOM] Categories satisfying Schoeder-Bernstein theorem
>> 
>> 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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>> FOM at cs.nyu.edu
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>> 
>> 
>> 
>

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