Lower Bagdomain as a Glueing

One of the important tools for working with the classifying toposes is Diaconescu's theorem, describing geometric morphisms to the topos of presheaves on a small category C in terms of at functors on C. In 4], this theorem is generalized from discrete categories to topological categories, with the additional requirement that the map assigning to an arrow its source is etale (a local homeomorphism). This generalization enables the author of 4] to interpret geometric morphisms to many new interesting toposes, e. g. those of sheaves on a simplicial space, in the sense of 1]. To deal with arbitrary Grothendieck toposes, we present a further generalization of Diaconescu's theorem, now from topological categories to \topical categories", i. e. internal categories in the category of to-poses and geometric morphisms. After extending suitably the notion of a principal bundle to this case, we will show that for certain such categories C, called here domain-etale, geometric morphisms to the classifying topos of C correspond to principal C-bundles. As in 4], this in particular gives an interpretation of geometric morphisms to a generalized glueing of a small diagram of toposes D : C ! Top in terms of D-augmented principal C-bundles. It is hoped that this interpretation can be used to construct representing objects for many interesting functors on toposes. In this note we present just one example of such an application of this generalized Diaconescu theorem. It concerns a particular description of the lower bagdomain topos from 3]: for a topos E, its lower bagdomain B L (E) will be expressed as glueing of a certain diagram naturally associated with E. Everywhere in the sequel, Top is the category of Grothendieck topo-ses, although most probably some of the toposes involved might be assumed unbounded. Morphisms of toposes mean geometric morphisms. When possible, we avoid mentioning 2-categorical aspects of our constructions ; e. g. pullback of toposes really means bi-pullback, Top-valued functors are really pseudofunctors, i. e. functors up to coherent canonical 2-isomorphisms, etc. 1