pushouts in toposes

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

Date: Mon, 16 Jun 1997 14:22:42 +0000
From: Cesc Rossello <dmifrl0@ps.uib.es>

Dear categorists

A PhD student of mine, Merce Llabres, and I we have got involved in
proving some properties of pushouts on toposes, similar (but somehow
dual) 
to those known for pullbacks. 
We are worried by the fact that perhaps somebody else has already proved
many of the 
results we are interested in.
So, before struggling to prove some probably well-known things, we
would really appreciate  some pointers to literature on the topic.

For instance:

Assume you have a family of pushouts in a topos
Ai ---> Bi
|            |
v  f       v
A------> B
 (all squares have the same bottom arrow    f)
with vertical arrows monic, and assume the pullbacks of (Ai -->A)_i
and (Bi---> B)_i exist, say A0 and B0, and consider the obvious square

A0 ---> B0
|              |
v  f         v
A------> B

It is a pushout square when the family of squares is finite, and 
for arbitrary families in all complete toposes we have tried 
(sets, hypergraphs, total unary algebras,
unary partial algebras with closed homomorphisms,...). 
Moreover, a proof (for complete toposes) can probably be derived from
the techniques in the paper by Kawahara in TCS vol 77 (1990). But, has 
somebody already proved (or disproved) such a result? 

Thanks in advance      Cesc Rossello