Re: pushouts in toposes

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

Date: Tue, 17 Jun 1997 10:32:21 -0400 (EDT)
From: Peter Freyd <pjf@saul.cis.upenn.edu>

Cesc Rossello asks about pushouts in topoi. In particular, assume that
for each i, 

   Ai ---> Bi      

   |       |

   A  ---> B
       f

is a pushout (same  f  each  i) and that the vertical maps are monic
(henceforth to be treated notationally as inclusion maps). Let  A0  be
the intersection of the  Ai's  and  B0  the intersection of the  Bi's.
Then is it the case that

   A0 ---> B0

   |       |

   A  ---> B   is also a pushout?

Yes for finite families, no for arbitrary families.

The case for finite families is an straightforward consequence of the
representation theorem for pre-topoi and the fact that such
representations preserve pushouts of monics. (See 1.636 and 1.65 in
Categories, Allegories.)

For the failure in the infinite-family case specialize to the case
that  f:A --> B  is also an inclusion map. The result, if true, would
translate to:

    A v /\Bi  =  /\(A v Bi).

Take sheaves on any non-discrete  T1-space,  X,  for a counterexample. 
Let  B  be the terminal sheaf (i.e.  X  itself), A  the complement of 
some non-isolated point and  {Bi}  the family of all other complements 
of one-element sets.