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!

Video of my talk on relative toposes

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

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!

Toposes online

We are pleased to announce the third (online) edition of the main international conference on topos theory, following the previous ones "Topos à l’IHES" and "Toposes in Como" :

For further information and registration, please visit the event website. We look forward to seeing many of you there!

Two talks on unification and morphogenesis

Next week I shall give two (video-recorded) talks, one in Italian and another one in English, on unification and morphogenesis from a topos-theoretic perspective:

27
April2021
Topoi come 'ponti’ unificanti: una morfogenesi matematicaIn questo seminario affronterò il tema dell’unificazione in matematica, e della dualità tra unità e molteplicità, dal punto di vista della teoria dei ‘ponti’ topos-teoretici. Discuterò in particolare il senso in cui il tipo di unificazione realizzato da questa teoria rappresenta una vera e propria ‘morfogenesi matematica’, intesa come insieme di dinamiche di differenziazione a partire da un’unità.

Slides available here.
10:30 (Italian time)Synergia Seminars,
University of Urbino
28
April2021
Unification and morphogenesis: a topos-theoretic perspectiveWe shall present some philosophical principles underlying the theory of toposes as unifying `bridges' in mathematics. More specifically, after reviewing the various types of unification which occur in mathematics, we shall discuss the way in which the connections established by topos-theoretic ‘bridges’ give rise to an authentic mathematical morphogenesis.   

Slides available here.
16:00 (French time) Seminario di Logica e Filosofia della Scienza,
University of Palermo

Many thanks to the organizers of these events, Proff. Pierluigi Graziani and Gianluigi Oliveri.


The over-topos at a model

I am pleased to announce the following paper, written in collaboration with Axel Osmond:

The over-topos at a model

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! 😉

On the "unifying notion" of topos

The video of my recent talk "Toposes as unifying spaces: historical aspects and prospects" at the Workshop in honor of Alexander Grothendieck’s legacy at the Universidad Nacional de Colombia is now available on YouTube:

This talk (whose slides can be downloaded here) discusses how the unifying concept of topos was introduced and conceived by Grothendieck, as well as the future prospects provided by the theory of toposes as ‘bridges’.

It is a shorter (and partially different) version of the lecture on the same subject I gave in 2018 at the ENS for the series Lectures Grothendieckiennes organized by Frédéric Jaëck (whose slides are available here):

In fact, I have recently finished writing (in French) my contribution for the Proceedings volume of that lecture series, which will be published by Spartacus: this paper can be downloaded here. An English translation is also in preparation.

Videos of my last research course

The videos of my recent research course on The geometry of morphisms and equivalences of toposes (whose slides can be downloaded here) are now available on YouTube:

Thanks again to John Alexander Cruz Morales for organizing this course, and for making the recordings available with the kind help of Javier Gutiérrez.