Artin glueing for quasitoposes

["Prof. Peter Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>]

Given a subobject of 1 in a topos, it's well known that one can
`split' the topos into complementary open and closed subtoposes,
and reconstruct the topos (up to equivalence) by applying Artin
glueing to these two subtoposes.

Hands up, all those of you who thought that the same thing works
for quasitoposes ... I thought so! It isn't true, as I have just
discovered.

Certainly, given a strong subobject U >--> 1 in a quasitopos E,
one can construct the `closed complement' of the open subquasitopos
E/U, in exactly the same way as one does for a topos: let's denote it
by C(U). It's also true that one has a `fringe functor' from E/U to
C(U), and that one gets a comparison functor from E to the
quasitopos obtained by glueing along this functor (again, the glueing
construction works for quasitoposes just as it does for toposes).
But, for this comparison to be an equivalence, one needs to know
that the inverse image functors E --> E/U and E --> C(U) are (not
just jointly faithful, but) jointly isomorphism-reflecting. And
that can fail: I have a counterexample in a slice of the quasitopos
of Frechet spaces (Elephant, A2.6.4(c).

Has anyone noticed this failure before? If so, has anyone actually
written it up?

Peter Johnstone