Récoltes et Semailles

The famous autobiographical work in two volumes by Alexander Grothendieck has just been officially published by Gallimard!

This book is a must for anyone who wants to learn about the vision inspiring Grothendieck's discoveries and reflect on ethical and sociological issues in mathematics and beyond.

I have been invited by the Editor to contribute a short text presenting the book, to be included in a booklet sold together with it and also containing texts by other mathematicians. Here is it:

Lectures Grothendieckiennes

The volume of proceedings of the lecture series Lectures Grothendieckiennes given in the academic year 2017-2018 at the Ecole Normale Superieure (Paris) is now available!

This book, which celebrates Grothendieck's mathematical heritage, is prefaced by Peter Scholze and features contributions by Pierre Cartier, Olivia Caramello, Alain Connes, Laurent Lafforgue, Colin McLarty, Gilles Pisier, Jean-Jacques Szczeciniarz and Fernando Zalamea.

My contribution is available here:

The videos of the lectures are available from YouTube at this link.

An article for "Science et Vie"

In the December issue of the French magazine Science et Vie there is an article of mine on applying the philosophy of 'bridges' and invariants in the context of social science:

Any comments are welcome.

Thanks again to the Chief Editor Thomas Cavaillé-Fol for inviting me to write this contribution!

A talk on proof-theoretic aspects of Grothendieck topologies

Tomorrow (Monday 22 November 2021, at 9:40 Central European Time) I will give a talk on "Deductive systems and Grothendieck topologies" for the Dagstuhl Seminar Geometric Logic, Constructivisation, and Automated Theorem Proving.

The abstract is as follows:

I will show that the classical proof system of geometric logic over a given geometric theory is equivalent to new proof systems based on the notion of Grothendieck topology. These equivalences result from a proof-theoretic interpretation of the duality between the quotients of a given geometric theory and the subtoposes of its classifying topos. Interestingly, these alternative proof systems turn out to be computationally better-behaved than the classical one for many purposes, as I will illustrate by discussing a few selected applications.

To attend the talk, you may click here.

A talk on relative topos theory

This evening, at 9pm CET, I will give an online talk on "Relative topos theory via stacks" at the University of Wisconsin Logic Seminar. Many thanks to the organizers of this Seminar, in particular to Prof. Steffen Lempp, for this invitation!

You may attend the talk as follows:

Zoom link to local UW logic seminar
Meeting ID: 986 3594 0882
Passcode: 003073

Abstract: In 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.

 

 

Relative topos theory via stacks

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):

Videos and slides of talks at "Toposes online"

The slides and videos of talks and courses given at the recent event Toposes online are now available from the conference website.

Thanks again to all the speakers at this event, who have allowed us to set up a very rich and varied programme, and to IHES for making the videos available on its YouTube channel!

Feel free to engage in discussions on the content of talks in the relevant section of the Around Toposes forum. Looking forward to seeing many of you there!

Welcome to "Toposes online"!

The school and conference “Toposes online” is about to start!

Thanks again to all our speakers for accepting our invitation or sending us a talk proposal. This has made it possible for us to set up a very rich and varied programme.

We are very happy to have attracted so many people (533 registered participants) to this event, coming from different areas of mathematics, physics, computer science, philosophy and engineering, and also from industry. This illustrates the wide-ranging impact of toposes, and the increasing interest in the subject, across different fields of knowledge.

We hope that this event will further stimulate interdisciplinary research inspired by topos-theoretic ideas, and its applications in different fields. The participants in the conference, as well as any other person interested in toposes, are warmly encouraged to join our community and exchange with each other through the Forum.

Many thanks to IHES and the University of Insubria for their support in organizing this event; the videos of all the talks and course lectures of "Toposes online" will be made available on the YouTube channel of IHES.

We look forward to seeing many of you online and on the forum!