Re: Constitutive Structures

[Richard Garner <richard.garner@mq.edu.au>]

Thanks Peter. It did occur to me last night that this probably was the
hyperconnected-localic factorisation and it is nice to have this
feeling confirmed! The problem is that the factorisation system I
described allows one to adjoin n-ary relations to _arbitrary_ objects
of Set[O], rather than merely to the generic object. In particular, as
you point out, of the first group of maps I listed it is only
necessary to consider the case n=1, and in fact on looking at at your
proof, orthogonality to this immediately implies orthogonality to the
last of the maps I listed.

Here's an attempt to overcome this; I suspect it will end up suffering
the same fate as the previous one but you never know! Rather than
describing a factorisation system on GTop, I am going to describe one
on GTop / Set[O]. The generating right maps will simply be the maps
from the classifying topos of an object equipped with an n-ary
relation into Set[O], though now these maps are viewed as maps over
Set[O]. If this generates a factorisation system (E, M), then its
M-maps with codomain E --> Set[O] will correspond to those things
constructible by repeatedly adjoining n-ary relations or equations
between n-ary relations to the specified object of E. Every such map
will be localic, but I think that the E-maps are no longer the
hyperconnected morphisms; the inverse image part of such a map need
only be full on subobjects of the specified object of its domain.

Now on factorising the unique map from R: Set --> Set[O] into the
terminal object of GTop / Set[O], it is possible that we obtain
something non-trivial which captures the structures (in geometric
logic) supported by the reals. I am however a bit hesitant about this
as my feeling is that if p: E --> F is an E-map of toposes over
Set[O], and F --> Set[O] is localic, then p probably is actually
hyperconnected (i.e., fullness on subobjects of the (image of) the
generic object implies fullness on all subobjects) so that we are back
in the situation we were in before...

Richard

On 16 April 2011 03:14, Prof. Peter Johnstone
<P.T.Johnstone@dpmms.cam.ac.uk> wrote:
> 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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