Videos of the Grothendieck Conference

The videos of the conference in honour of Grothendieck held in May at Chapman University are now available from Youtube.

My own talk at the conference was entitled “The unifying ‘notion’ of topos“:

The slides can be downloaded here. A written text of my presentation will be available soon.

A course on toposes as ‘bridges’

In the next weeks, starting from this Friday at 14:00 London time, Laurent Lafforgue will give an online course on toposes as ‘bridges’ at the University of Warwick.

The programme is as follows:

Lecture I: Grothendieck topologies, sheaves, toposes and points.

Date: 18 February 2022 (Friday) at 14:00-15:00 (London time).

Abstract: The purpose of this first lecture will be to introduce the notion of Grothendieck topology on a category, the associated notions of sheaves and toposes, and the derived notion of point of a topos. A special attention will be given to the problem of generation of Grothendieck topologies which, as will become clear in the third lecture, is of great significance and magnitude. Comparing the representations of objects of an arbitrary topos in terms of a presenting site and their evaluations at points provides some extremely general form of non-linear Fourier decompositions and Fourier transforms.


Lecture II: Linguistic descriptions of points and first-order geometric theories.

Date: 25 February 2022 (Friday) at 14:00-15:00 (London time).

Abstract: It will be shown how any presentation of a Grothendieck topos by a site allows to give a linguistic description of set-based points and generalised points of a topos. These descriptions make up bridges from geometry to languages, i.e. to words and grammar rules. They also provide a good way to introduce the general notions of first-order languages and first-order geometric theories.


Lecture III: Classifying toposes, toposes as bridges and the equivalence between first-order provability and generation of Grothendieck topologies.

Date: 4 March 2022 (Friday) at 14:00-15:00 (London time).

Abstract: It will be shown that any geometric first-order theory defines a “functor of models” which associates to any topos the category of the models of this theory with coefficients in this topos, and that this functor of models is always representable by a so-called “classifying topos”: it is characterized by the property that models of the theory identify with points of this associated classifying topos. The theory of classifying toposes, which was developped by W. Lawvere and the school of categorical logic in the 1970’s, building on some seminal ideas of Grothendieck, was given new impetus with the technique of “toposes as bridges” introduced by O. Caramello in her 2009 PhD thesis. This technique consists in exploiting the fact that any topos can be represented by a double infinite diversity of presenting sites and of geometric theories, in order to develop a general theory of relations between mathematical theories. A particular example of that is the interpretation in terms of quotient theories of the invariant of toposes consisting in their ordered sets of subtoposes. It provides an equivalence between the general problem of provability in the context of first-order geometric theories and the problem of generation of Grothendieck topologies on small categories.

For more information, including the links to attend the lectures, please visit the course webpage.

Ontologies and toposes

The slides of the double talk given by Laurent Lafforgue and myself at the recent Huawei workshop on semantics are now available:

Our talk is based on an expository article written by AI theorists on the general theme of “ontology” and “knowledge representation”. We explain that the theory of Grothendieck toposes, of their geometric presentations by sites and of their linguistic descriptions in terms of first-order theories provides the means to incarnate in mathematical objects amenable to computations what could have seemed at first sight loose philosophical views. We present a number of important basic ideas about Grothendieck toposes and motivations for beginning to study them, and give an idea of the possibilities of the theory of “toposes as bridges” and its significance for the problem of “knowledge representation”.

Working group on proofs and Grothendieck topologies

A working group of about 20 researchers has formed to investigate computational aspects of the methodology ‘toposes as bridges’, with particular reference to the proof-theoretic equivalences established in Chapters 3 and 8 of my book Theories, Sites, Toposes: Relating and studying mathematical theories through topos-theoretic ‘bridges’ and described in these slides.

Bridge between Grothendieck topologies and quotients

The organizer is Laurent Lafforgue, and the meetings will take place at the Huawei Lagrange Center for Mathematics and Computation in Paris, starting from the first one, which has happened today.

A final goal is to implement these techniques on a computer, to automatically generate mathematical results by exploiting the capacity of ‘bridges’ to significantly transform the level of complexity of notions and results. Back in 2010, when I first evoked this possibility in the paper The unification of Mathematics via Topos Theory, that idea was regarded with a lot of skepticism, as something almost too good to be true. Now, the time is ripe to start making that dream into reality.

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):

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! 😉