Date: Mon, 18 Nov 1996 17:10:41 -0400 (AST)
Subject: Separated objects

Date: Mon, 18 Nov 1996 17:48:41 +0100 (MET)
From: Sebastiano Vigna <vigna@dotto.usr.dsi.unimi.it>

I need to find out what is known about the separated objects of an
elementary (or even Grothendieck) topos. Pointers to the relevant
literature would be immensely appreciated.

                                                seba

Date: Tue, 19 Nov 1996 08:23:31 -0400 (AST)
Subject: Re: Separated objects

Date: Tue, 19 Nov 96 09:48 GMT
From: Dr. P.T. Johnstone <P.T.Johnstone@pmms.cam.ac.uk>

Assuming you mean the separated objects for a Lawvere--Tierney local
operator, what is known (and essentially all that is known) is that
they form a quasitopos. This result is due to me (rather to my
surprise): the first proof of it is in my paper "On a topological
topos" in Proc. London Math. Soc. 38 (1979), 237--271. There is a
lot more detail in Oswald Wyler's book "Lecture Notes on Topoi and
Quasitopoi" (World Scientific, 1991). It remains an open problem
(and a very hard one, I think) whether every quasitopos is
representable as the separated objects for some local operator on
a topos.

Peter Johnstone