Re: Artin glueing for quasitoposes

Re: Artin glueing for quasitoposes

[Prof. Peter Johnstone]

Here are the details of the counterexample mentioned in my earlier
e-mail, for anyone who wants to see them.

Let E be the quasitopos of Frechet spaces (called subsequential
spaces in my paper "On a topological topos"): all you need to
know about this category is that it contains the category S of
sequential spaces as a full subcategory, closed under limits
(not under all colimits, but in fact all colimits that occur in
the following discussion are preserved). All the action takes place
inside S. Let N be the discrete space of natural numbers, and
N+ its one-point compactification N \cup \{\infty\}. Let A be
the space obtained from the disjoint union of two copies of N+
by identifying the two copies of \infty, and let A' be the disjoint
union of N and N+. Clearly, there is a morphism A' \to A which
is bijective on points (hence, both monic and epic) but not an
isomorphism.

However, if we regard A and A' as spaces over N+ in the obvious
way, the morphism A' \to A becomes an isomorphism when we pull
it back along the inclusion N \to N+, and also when we form the
pushouts of

           A x  N -------> A(')
              N+
             |
             |
             v
             N

since both such pushouts are isomorphic to N+. This says that,
if we work in the quasitopos E/N+, the morphism A' \to A is mapped to
an isomorphism in both the open subquasitopos E/N and its closed
"complement".

I should say that I came to consider this question as a result of
a seminar talk today by Pawel Sobocinski, which raised the question
of whether quasitoposes are quasi-adhesive categories. This example
shows that the quasitopos of Frechet spaces is not quasi-adhesive.

Peter Johnstone

Artin glueing for quasitoposes

[Prof. Peter Johnstone]

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