Toposes Online - 24-30 June 2021 - minicourses and conference IHES
Topos theory can be regarded as a unifying subject within Mathematics; in the words of Grothendieck, who invented the concept of topos, “It is the theme of toposes which is this “bed”, or this “deep river”, in which come to be married geometry and algebra, topology and arithmetic, mathematical logic and categorytheory, the world of the continuous and that of the “discontinuous” or “discrete” structures. It is what I have conceived of most broad, to perceive with finesse, by the same language rich of geometric resonances, an “essence” which is common to situations most distant from each other”.
The format of the event is the same as that of the other two editions: it will consist of a three-day school, offering introductory courses for the benefit of students and mathematicians who are not already familiar with topos theory, followed by a three-day congress featuring both invited and contributed presentations on new theoretical advances in the subject as well as applications of toposes in different fields such as algebra, topology, number theory, algebraic geometry, logic, homotopy theory, functional analysis, and computer science.
The main aim of this conference series is to celebrate the unifying power and interdisciplinary applications of toposes and encourage further developments in this spirit, by promoting exchanges amongst researchers in different branches of mathematics who use toposes in their work and by introducing a new generation of scholars to the subject.
Because of the pandemic, this edition of the conference will take place entirely online. The participants may take advantage of the associated forum to discuss with each other (please register to it if you wish to post messages).
- Olivia Caramello (University of Insubria and IHES)
- Laurent Lafforgue (IHES)
- Charles Rezk (University of Illinois)
- Samson Abramsky (University of Oxford)
- Jean-Claude Belfiore (Huawei)
- Daniel Bennequin (University of Paris 7)
- Dustin Clausen (University of Copenhagen)
- Jens Hemelaer (University of Antwerp)
- Luca Prelli (University of Padua)
- Peter Scholze (University of Bonn)
- Ivan Tomasic (Queen Mary University of London)
Scientific and Organizing Committee
- Olivia Caramello
- Alain Connes
- Laurent Lafforgue
We gratefully acknowledge IHES and the University of Insubria for their support; in particular, the videos of "Toposes online" will be made available through the IHES YouTube channel.
- Olivia Caramello: "Introduction to sheaves, stacks and relative toposes" VIDEO 1, VIDEO 2, VIDEO 3, VIDEO 4
Abstract: This course provides a geometric introduction to (relative) topos theory.
The first part of the course will describe the basic theory of sheaves on a site, the main structural properties of Grothendieck toposes and the way in which morphisms between toposes are induced by suitable kinds of functors between sites.
The second part, based on joint work with Riccardo Zanfa, will present an approach to relative topos theory (i.e. topos theory over an arbitrary base topos) based on stacks and a suitable notion of relative site.
- Laurent Lafforgue: "Classifying toposes of geometric theories" VIDEO 1, VIDEO 2, VIDEO 3, VIDEO 4
Abstract: The purpose of these lectures will be to present the theory of classifying toposes of geometric theories. This theory was developped in the 1970's by Lawvere, Makkai, Reyes, Joyal and other catagory theorists, systematising some constructions of Grothendieck and his student Monique Hakim, but it still deserves to be much better known that it actually is.
The last part of the lectures will present new developpments due to Olivia Caramello which, based on her principle of "toposes as bridges", make the theory of classifying toposes more applicable to concrete mathematical situations : in particular, the equivalence between geometric provability and computing on Grothendieck topologies, and general criteria for a theory to be of presheaf type.
- Charlez Rezk: "Higher Topos Theory" VIDEO 1, VIDEO 2, VIDEO 3, VIDEO 4
Abstract: In this series of lectures I will give an introduction to the concept of "infinity topoi", which is an analog of the notion of a "Grothendieck topos" which is not an ordinary category, but rather is an "infinity category".
No prior knowledge of higher category theory will be assumed.
- Samson Abramsky: "The sheaf-theoretic structure of contextuality and non-locality" VIDEO
Abstract: Quantum mechanics implies a fundamentally non-classical picture of the physical world. This non-classicality is expressed in it sharpest form in the phenomena of non-locality and contextuality, articulated in the Bell and Kochen-Specker theorems. Apart from the foundational significance of these ideas, they play a central role in the emerging field of quantum computing and information processing, where these non-classical features of quantum mechanics are used to obtain quantum advantage over classical computational models. The mathematical structure of contextuality, with non-locality as a special case, is fundamentally sheaf-theoretic. The non-existence of classical explanations for quantum phenomena corresponds precisely to the non-existence of certain global sections. This leads to both logical and topological descriptions of these phenomena, very much in the spirit of topos theory.
This allows the standard constructions which witness these results, such as Kochen-Specker paradoxes, the GHZ construction, Hardy paradoxes, etc., to be visualised as discrete bundles. The non-classicality appears as a logical twisting of these bundles, related to classical logical paradoxes, and witnessed by the non-vanishing of cohomological sheaf invariants. In this setting, a general account can be given of Bell inequalities in terms of logical consistency conditions. A notion of simulation between different experimental situations yields a category of empirical models, which can be used to classify the expressive power of contextuality as a resource. Both quantitative and qualitative, and discrete and continuous features arise naturally.
- Jean-Claude Belfiore: "Beyond the statistical perspective on deep learning, the toposic point of view: Invariance and semantic information" (joint work with Daniel Bennequin) VIDEO
Abstract: The last decade has witnessed an experimental revolution in data science and machine learning, essentially based on two ingredients: representation (or feature learning) and backpropagation. Moreover the analysis of the behavior of deep learning is essentially done through the prism of probabilities. As long as artificial neural networks only capture statistical correlations between data and the tasks/questions that have to be performed/answered, this analysis may be enough. Unfortunately, when we aim at designing neural networks that behave more like animal brains or even humans’ ones, statistics is not enough and we need to perform another type of analysis. By introducing languages and theories in this framework, we will show that the problem of learning is, first, a problem of adequacy between data and the theories that are expressed. This adequacy will be rephrased in terms of toposes. We will unveil the relation between the so-called “generalization” and a stack that models this adequacy between data and the tasks.
Finally a five level perspective of learning with neural networks will be given that is based on the architecture (base site), a presemantic (fibration), languages, theories and the notion of semantic information.
- Daniel Bennequin: "Topos, stacks, semantic information and artificial neural networks" (joint work with Jean-Claude Belfiore) VIDEO
Abstract: Every known artificial deep neural network (DNN) corresponds to an object in a canonical Grothendieck’s topos; its learning dynamic corresponds to a flow of morphisms in this topos. Invariance structures in the layers (like CNNs or LSTMs) correspond to Giraud’s stacks. This invariance is supposed to be responsible of the generalization property, that is extrapolation from learning data under constraints. The fibers represent pre-semantic categories (Culioli, Thom), over which artificial languages are defined, with internal logics, intuitionist, classical or linear (Girard). Semantic functioning of a network is its ability to express theories in such a language for answering questions in output about input data. Quantities and spaces of semantic information are defined by analogy with the homological interpretation of Shannon’s entropy (P.Baudot and D.B.). They generalize the measures found by Carnap and Bar-Hillel (1952). Amazingly, the above semantical structures are classified by geometric fibrant objects in a closed model category of Quillen, then they give rise to homotopical invariants of DNNs and of their semantic functioning. Intentional type theories (Martin-Löf) organize these objects and fibrations between them. Information contents and exchanges are analyzed by Grothendieck’s derivators.
- Dustin Clausen: "Toposes generated by compact projectives, and the example of condensed sets" VIDEO
Abstract: The simplest kind of Grothendieck topology is the one with only trivial covering sieves, where the associated topos is equal to the presheaf topos. The next simplest topology has coverings given by finite disjoint unions. From an intrinsic perspective, the toposes which arise from such a topology are exactly those which, as a category, have the useful property that they are generated by compact projective objects. I will discuss some general aspects of this situation, then specialize to a specific example, that of condensed sets. This is joint work with Peter Scholze.
- Jens Hemelaer: "Toposes of presheaves on monoids as generalized topological spaces" VIDEO
Abstract: Various ideas from topology have been generalized to toposes, for example surjections and inclusions, local homeomorphisms, or the fundamental group. Another interesting concept, that is less well-known, is the notion of a complete spread, that was brought from topology to topos theory by Bunge and Funk. We will discuss these concepts in the special case of toposes of presheaves on monoids. The aim is to gain geometric intuition about things that are usually thought of as algebraic.
Special attention will go to the underlying topos of the Arithmetic Site by Connes and Consani, corresponding to the monoid of nonzero natural numbers under multiplication. The topological concepts mentioned earlier will be illustrated using this topos and some of its generalizations corresponding to maximal orders.The talk will be based on joint work with Morgan Rogers and joint work with Aurélien Sagnier.
- Luca Prelli: "Sheaves on T-topologies" VIDEO
Abstract: Let T be a suitable family of open subsets of a topological space X stable under unions and intersections. Starting from T we construct a (Grothendieck) topology on X and we consider the associated category of sheaves. This gives a unifying description of various constructions in different fields of mathematics.
- Peter Scholze: "Liquid vector spaces" VIDEO
Abstract: (joint with Dustin Clausen) Based on the condensed formalism, we propose new foundations for real functional analysis, replacing complete locally convex vector spaces with a variant of so-called p-liquid condensed real vector spaces, with excellent categorical properties; in particular they form an abelian category stable under extensions. It is a classical phenomenon that local convexity is not stable under extensions, so one has to allow non-convex spaces in the theory, and p-liquidity is related to p-convexity, where 0<p<=1 is an auxiliary parameter. Strangely, the proof that the theory of p-liquid vector spaces has the desired good properties proceeds by proving a generalization over a ring of arithmetic Laurent series.
- Ivan Tomasic:"A topos-theoretic view of difference algebra"
Abstract: Difference algebra was founded by Ritt in the 1930s as the study of rings and modules with distinguished endomorphisms thought of as `difference operators’. Aiming to introduce cohomological methods into the subject, we view difference algebra as the study of algebraic objects in the topos of BN of difference sets, i.e., actions of the additive monoid N of natural numbers. Guided by the general principle that the G-equivariant algebraic geometry (where G is a group, monoid, groupoid or a category) should correspond to the relative algebraic geometry over the base topos BG, we develop difference algebraic geometry as relative algebraic geometry over the base topos BN. We extend the framework of Hakim’s 1970s monograph to include the theories of the fundamental group and the \’etale cohomology of relative schemes over a general base topos, and derive consequences in the difference case.
- Peter Arndt: “Ranges of functors and geometric classes via topos theory” VIDEO
- Georg Biedermann (joint work with Mathieu Anel, Eric Finster, and André Joyal): “Higher Sheaves" VIDEO
- Ivan Di Liberti: “Towards higher topology” VIDEO
Francesco Genovese (joint work with Julia Ramos González): “A derived Gabriel-Popescu Theorem for T-structures via derived injectives” VIDEO
Matthias Hutzler: “Gluing classifying toposes” VIDEO
- Ming Ng (joint work with Steve Vickers): “Adelic Geometry via Topos Theory” VIDEO
- Rasekh Nima: “Every Elementary Higher Topos has a Natural Number Object”
- Axel Osmond (joint work with Olivia Caramello): “The over-topos at a model” VIDEO
Jason Parker: “Covariant Isotropy of Grothendieck Toposes” VIDEO
Morgan Rogers: “Toposes of Topological Monoid Actions” VIDEO