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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