The paper carries out a systematic investigation of the functors between sites which induce morphisms between relative toposes.
It culminates in a generalisation of Diaconescu’s theorem for relative toposes, formulated in the language of fibrations and stacks according to the foundations for relative topos theory introduced in the paper Relative topos theory via stacks.
The paper introduces, the context of relative topos theory via stacks, the new notions of existential fibred site and of existential topos of such a site. These concepts allow us to develop relative topos theory in a way which naturally generalizes the construction of toposes of sheaves on locales and also provides a framework for investigating the connections between Grothendieck toposes as built from sites and elementary toposes as built from triposes.
The paper also contains a fibred generalisation of the ideal-completion of a preorder site, a construction which has played a key role in the development of formal topology since the eighties:
We expect this construction to find several applications, in particular in connection with the generation of dualities from multiple representations of toposes in terms of fibred preorder sites (in the spirit of this paper), but also in Logic; indeed, applications to the construction of completions of doctrines will be provided in a forthcoming paper by my doctoral student Joshua Wrigley.
Lastly, the paper provides an explicit description of the hyperconnected-localic factorization of a geometric morphism in terms of internal locales, with applications to the construction of alternative syntactic sites for the classifying topos of a theory.
I take this opportunity to send to all of you my Season’s Greetins and very best wishes for the New Year!
UPDATE: Slides presenting the contents of the paper are available here.