The slides of my introductory course to relative toposes recently given at the Lagrange Center are now available:
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.
Next week, on the 8th and 9th of September, the Ninth Symposium on Compositional Structures (SYCO 9) is taking place in Como.
I have been invited to give a talk on this occasion. The title of my presentation is Relative toposes as a generalization of locales. The abstract is as follows:
The aim of this talk is to present a way for representing relative toposes which naturally generalizes the construction of the topos of sheaves on a locale, and which is particularly effective for describing the morphisms between relative toposes in a concrete way. Our theoretical framework is based on the language of stacks and fibred sites, and provides, amongst other things, a unified setting for investigating the relationships between Grothendieck toposes as built from sites and elementary toposes as built from triposes.
Looking forward to seeing many of you in Como!
I am glad to announce the first version of our joint work with Riccardo Zanfa on relative toposes:
Here is the abstract:
We introduce new foundations for relative topos theory based on stacks. One of the central results in our theory is an adjunction between the category of toposes over the topos of sheaves on a given site (C,J) and that of C-indexed categories. This represents a wide generalization of the classical adjunction between presheaves on a topological space and bundles over it, and allows one to interpret several constructions on sheaves and stacks in a geometrical way; in particular, it leads to fibrational descriptions of direct and inverse images of sheaves and stacks, as well as to a geometric understanding of the sheafification process. It also naturally allows one to regard any Grothendieck topos as a ‘petit’ topos associated with a ‘gros’ topos, thereby providing an answer to a problem posed by Grothendieck in the seventies. Another key ingredient in our theory is a notion of relative site, which allows one to represent arbitrary geometric morphisms towards a fixed base topos of sheaves on a site as structure morphisms induced by relative sites over that site.
We shall progressively release expanded versions of this text contaning new developments in the directions sketched in the introduction.
This work has been recently presented at the conference Toposes online:
The following video focuses on the specialization of the fundamental adjunction in the setting of presheaves (or discrete fibrations):
The video of my yesteday talk on relative toposes is already available from YouTube:
Thanks again to the organizers for their invitation, and for making the recording available so quickly!
Next Tuesday I shall give a talk for ItaCa Fest 2021 on my work in progress with Riccardo Zanfa providing new foundations for relative topos theory based on stacks:
Here is the Zoom link to attend the talk: https://zoom.us/j/94880770089?pwd=clgxK2VkVEE5Ymw5ME1QWktiWExUZz09
Thanks to the ItaCa Fest organizers for this invitation, and looking forward to seeing many of you there!
I am pleased to announce the following paper, written in collaboration with Axel Osmond:
This work introduces a new topos-theoretic construction, that of the over-topos at a model of a geometric theory in a Grothendieck topos, and investigates both its logical and geometric aspects. Here is the abstract:
With a model of a geometric theory in an arbitrary topos, we associate a site obtained by endowing a category of generalized elements of the model with a Grothendieck topology, which we call the antecedent topology. Then we show that the associated sheaf topos, which we call the over-topos at the given model, admits a canonical totally connected morphism to the given base topos and satisfies a universal property generalizing that of the colocalization of a topos at a point. We first treat the case of the base topos of sets, where global elements are sufficient to describe our site of definition; in this context, we also introduce a geometric theory classified by the over-topos, whose models can be identified with the model homomorphisms towards the (internalizations of the) model. Then we formulate and prove the general statement over an arbitrary topos, which involves the stack of generalized elements of the model. Lastly, we investigate the geometric and 2-categorical aspects of the over-topos construction, exhibiting it as a bilimit in the bicategory of Grothendieck toposes.
The construction of the over-topos can also be dualized, providing a wide generalization of Grothendieck-Verdier’s notion of localization of a topos at a point.
This paper combines a variety fo techniques and touches several distinct themes, introducing new ideas or constructions in connection with each of them :
- Syntactic categories and classifying toposes
- Totally connected toposes and colocalizations
- Grothendieck topologies on fibrations
- Computation of Grothendieck topologies generated by different families of sieves
- Geometric morphisms and stacks associated with them
- Giraud’s construction of the classifying topos of a stack
- 2-categorical constructions in the bicategory of Grothendieck toposes
In my forthcoming joint work with Riccardo Zanfa we shall introduce a whole new framework for developing relative topos theroy via stacks, thereby providing a broad context where the results of this paper can be understood. Stay tuned! 😉