Re: pushouts in toposes

[categories <cat-dist@mta.ca>]

Date: Tue, 17 Jun 1997 11:54:07 -0400 (EDT)
From: Peter Freyd <pjf@saul.cis.upenn.edu>


The distributivity condition is not only necessary and but a
sufficient condition for the pushout result. A square

   A'--> B'

   |     |

   A --> B
      f

in which the vertical arrows are monic is a pushout iff the
following three conditions hold:

  1)  the square is a pullback;
  2)  B  is the union of  Image(f)  and  B';
  3)  the congruence, E, induced by  f  is the union of the
      identity relation and  E ^ (A' x A').

Hence, the desired result reduces to:

   B  =  Image(f) v /\Bi;
   E  =  I v /\(E ^ (Ai x Ai))  =  I v (E ^ /\(Ai x Ai)).