Fibred sites and existential toposes

I have just uploaded to the ArXiv a new paper, entitled “Fibred sites and existential toposes“.

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.

On Copernicus and Grothendieck

The slides of my recent talk at the conference in honor of Jean-Jacques Szceciniarz are available here:

The talk presents an analogy between the role of the Sun in Copernicus’ vision and that of Grothendieck toposes as ‘bridges’ between different mathematical theories, by building on the general notion of a ‘bridge’ object:

The Sun as a ‘bridge’ object

Interview for Radio3 Scienza

Today’s episode of the Italian national radio Radio3 Scienza was entirely devoted to the newly inaugurated Grothendieck Institute. I was interviewed by Roberta Fulci together with Laurent Lafforgue and Johanna Grothendieck, who both participated in the inaugural event on Saturday.

You may listen to the recording here.