Re: Constitutive Structures

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

Dear Richard,

That's an ingenious idea, but I don't think it helps. The
factorization system is indeed a well-known one: it's the
hyperconnected--localic factorization [proof below], and
it is indeed true that M-maps into Set[O] correspond to
single-sorted geometric theories (Elephant, D3.2.5). But
every morphism Set --> Set[O] (in particular the one which
classifies the real numbers) is localic, so you just end up
with the topos of sets.

Here's the proof. The morphisms you describe are all localic,
so it's enough to prove that any morphism orthogonal to them all
is hyperconnected. But orthogonality to the last morphism you
list, for a morphism f: F --> E, says precisely that if m is
a mono in E and f^*(m) is iso then m is iso, i.e. that f is
surjective. Then orthogonality to the first group (actually
you only need the case n=1) says that f^* is `full on subobjects',
i.e. that every subobject of f^*(A) is of the form f^*(B) for a
unique (up to isomorphism) B >--> A. Applying this to the graphs
of morphisms, you get that f^* is full in the usual sense;
applying it to arbitrary subobjects, you get the criterion for
hyperconnectedness given in Elephant, A4.6.6(ii).

Peter Johnstone

On Fri, 15 Apr 2011, Richard Garner wrote:

> Here's a possible answer using toposes. I don't really know enough
> topos theory to do this properly so I will be busking it a bit;
> hopefully someone more knowledgeable than I can tell me what I am up
> to! We define a factorisation system (E,M) on the 2-category of
> Grothendieck toposes, generated by the following M-maps. For each n,
> we take the obvious geometric morphism from the classifying topos of
> an object equipped with an n-ary relation to the object classifier;
> and we take that geometric morphism from the object classifier to the
> classifying topos of a monomorphism which classifies the identity map
> on the generic object. With any luck this generates a factorisation
> system on GTop; with equal luck it is a well-known one, but my
> knowledge of the taxonomy of classes of geometric morphisms is
> sufficiently hazy that I cannot say which it might be. In any case,
> the hope is that M-maps into the object classifier should correspond
> to single-sorted geometric theories. Now we work in the category of
> such M-maps into Set[O], and in there, there is an object which
> represents all the constitutive substructures of the reals. The object
> in question is obtained as the M-part of the (E,M) factorisation of
> the geometric morphism Set -> Set[O] which classifies the real
> numbers; it is the "complete theory of the reals", but not with
> respect to any particular structure, but rather with respect to all
> possible structures (within geometric logic) that we might impose on
> it. Unfortunately this would not capture, e.g., the second-order
> structures we might impose on the reals, but it's a start.
>
> (Of course, if we were merely interested in structures expressible by
> finitary algebraic theories, then we could consider the category of
> finitary monads on Set, and in there, the finitary coreflection of the
> codensity monad of the reals. That was my initial reaction to this
> problem, and the above is supposed to generalise this in some sense).
>
> Richard
>

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