Re: Constitutive Structures

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

In fact, I think the condition of being an E-map in this new sense
says: f : E --> F over Set[O] is such a map just when, on taking the
hyperconnected-localic factorisations E --> E' --> Set[O] and F --> F'
--> Set[O], the induced geometric morphism f' : E' --> F' is
hyperconnected. In particular, if F --> Set[O] is localic, then f is
an E-map if and only if it is hyperconnected. So this gets us no
further than before. Oh well!

Richard

On 16 April 2011 10:31, Richard Garner <richard.garner@mq.edu.au> wrote:
> 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
>

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