Tag Archives: Stacks

Visit to Lyon

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On Tuesday the 5th of March I will give a talk at the ENS de Lyon, in the framework of their Seminar “Logique, Mathématiques, Informatique, Raisonnement“, about the work on relative topos theory that I have been pursuing in the last years, starting with the paper Relative topos theory via stacks (with R. Zanfa), and which I am currently developing with my Ph.D. students at the Grothendieck Institute.

The seminar will take place at 18 in the Amphi A / Grande Nef 1-2 of the Campus Monod of the ENS de Lyon; you may download the poster here.

Looking forward to it!

A generalisation of Diaconescu’s theorem for relative toposes

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A new paper, written jointly with my Ph.D. student Léo Bartoli, is available from the ArXiv:

On morphisms of relative toposes

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.

Slides: “Relative topos theory via stacks: an introduction”

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The slides of my introductory course to relative toposes recently given at the Lagrange Center are now available:

RelativeToposTheoryViaStacksAnIntroductionDownload

Slides: “Toposes as ‘bridges’ for mathematics and artificial intelligence”

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The slides of my talk at the Workshop on Semantic Information and Communication are available for download:

ToposesForMathsAndAISemanticsWorkshopDownload

Towards a semantic 6G

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I am happy to give a talk tomorrow at the “Workshop on Semantic Information and Communication” held at the Lagrange Mathematics and Computing Research Center in Paris.

The workshop, which takes place on the 7th and 8th of March, is entitled “Towards a semantic 6G” and gathers experts from different fields whose research can contribute to the development of new conceptual architectures for communication based on semantic information.

As I shall explain in my talk, topos theory is relevant for the goals of semantic communication, and AI techniques based on semantic information more generally, in several different ways. In particular, given the capacity of toposes of embodying the semantics of a great variety of situations, using toposes as ‘bridges’ enables effective transfers of knowledge across different ways of expressing information (formalized as different ways of presenting toposes):

Semantics as a ‘bridge’

The links to attend the event are available from the conference webpage.

Looking forward to it!

Fibred sites and existential toposes

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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.

SYCO 9 in Como

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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!

Relative topos theory via stacks

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I am glad to announce the first version of our joint work with Riccardo Zanfa on relative toposes:

Relative topos theory via stacks

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):

Video of my talk on relative toposes

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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!

A talk on relative toposes

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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:

15
June2021
Relative topos theory via stacksIn this talk, based on joint work with Riccardo Zanfa, we shall 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 (relatively small) 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.
14:30 (Italian time)ItaCa Fest 2021

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!