School courses

Marc Aiguier (CentraleSupélec, Université Paris-Saclay)

Logic, categories and topos: from category theory to categorical logic

Mathematical logic as the foundation of mathematics was first studied in the set-theoretic framework. Following Grothendieck’s works on topos in the late 1950s, and the fact that the latter possessed properties that brought them closer to sets, logicians under the leadership of Lawvere proposed to extend the semantics of first-order logic and its fragments (higher-order logic, geometric logic, etc.) to category theory and among other things to a finitary axiomatization of topos, elementary topos.

The outline of this course will be as follows:

• Syntax of logics (signature, terms, formulas, sequent and theory)
• Categorical semantics (subobject, structures and evaluation of terms)
• Satisfaction of FO formulas in Heyting categories
  • Lawere’s hyperdoctrines
  • Elementary toposes
  • Interpretation of sequents
  • Kripke-Joyal semantics
• Interpretation of HOL in elementary toposes
• Geometric logic
  • Geometric category
  • Grothendieck toposes
• Inference systems (rules, correctness and completeness)
  • Syntactic category and universal model
  • Representable functor T-Mod
• Classifying topos

Olivia Caramello (University of Insubria and Grothendieck Institute)

Relative toposes for the working mathematician

Relativity techniques for schemes have played a key role in Grothendieck’s refoundation of algebraic geometry. We shall give an introduction to the relativity techniques for toposes, formulated in the language of stacks and fibrations, that we have been developing since 2020.

After recalling the necessary preliminaries, we will discuss the problem of representing geometric morphisms both in terms of morphisms and comorphisms of sites; this will lead to the notion of “relative site” and the study of the associated “relative toposes”.

We will show that the usual presheaf-bundle adjunction for topological spaces generalizes to arbitrary sites, by replacing continuous maps to the space with relative toposes (with respect to the topos of sheaves on the given site).

We shall then present a generalisation of Diaconescu’s equivalence for relative toposes, formulated in the language of relative sites.

Lastly, we will introduce the concept of “existential fibred site” (a broad, site-theoretic generalisation of the notion of (hyper)doctrine), and that of “existential topos” of such a site: these notions notably allow us to develop relative topos theory in a way which naturally generalizes the construction of toposes of sheaves on locales, providing a unified framework for investigating the connections between Grothendieck toposes as built from sites and elementary toposes as built from triposes.

We shall discuss several examples and applications of the notions and results presented in the lectures, of algebraic, geometric and logical nature.

Part of the material presented in the course comes from joint works with R. Zanfa, L. Bartoli and R. Lamagna.

Alain Connes (IHES)

Knots, Primes, and the Scaling Site

The topic of my course will focus on the interrelation between a very specific topos, the scaling site, and the well-known analogy going back to 1963 between knots and prime numbers.

The topos, which is the scaling site, is simple to define as the semi-direct product of the half-line of positive or zero real numbers by the action of the positive integers, by multiplication.

It turned out in 2014 (in joint work with C. Consani) that the points of this topos were identified with a space which had occurred in 1996 in non-commutative geometry related to the Riemann zeta function, and which is obtained by starting with the adele class space and dividing it by the action of the maximal compact subgroup of the Idele class group.

The main point of the class will be to understand the use of topos theory and of non-commutative geometry in order to understand the scaling site and the finite abelian covers of the scaling site associated to finite abelian extensions of the field of rational numbers.

Grothendieck, by his theory of the etale fundamental group, has extended Galois theory from the context of fields to the context of schemes.

The main new result which I will present (a joint work with C. Consani) is that the scaling site and the adele class space allow one to extend the class field theory isomorphisms, which usually relate Galois groups with groups of adelic nature, to the situation where one is no longer starting with a Galois group, but one is starting with a scheme intimately related to the field of rational numbers, and one associates to this scheme its class field theory counterpart, which can be seen either at the adelic level or at the topos level.

Denis-Charles Cisinski (University of Regensburg)

Synthetic ∞-category theory and elementary ∞-topoi

We will propose a formal language of category theory, independent of set-theoretic foundations. Formally, this language is a variation of type theory, but, as for set theory, there is a “naïve” version in natural language (with which we actually work) and this is what we will introduce.

Although this language looks like (and arguably can be) the one of ordinary category theory, a suitable version of Voevodsky’s univalence axiom will turn it into the language of infinity-categories as developed by Joyal and Lurie, in a version that is expressive enough to prove all the basic results of (higher) category theory – including the theory of Grothendieck topoi.

This language also expresses (higher) category theory internally to any (higher) topos.

The goal of this lecture series is to introduce such a language and to explain why it is in fact a way to define what is an elementary topos in the setting of higher category theory.

Dennis Gaitsgory (Max Planck Institute for Mathematics)

2-Fourier-Mukai transform and geometric Langlands for non-constant group-schemes

 Let  be a curve, and  a semispmple group. Let  be the moduli stack of -bundles on . Let be the Langlands-dual of , and let  be the stack of -local systems on . The geometric Langlands conjecture (now a theorem) says that there is an equivalence

(*) .

 (We suppress the difference between  and  as it will play no role in what follows.)

 Now, one can twist the two sides of (*) as follows:

 (a) Using the short exact sequence

, given a gerbe  on with respect to ,

we can form a twist  of  as a group-scheme over , and consider the category 

(b) Using the short exact sequence

, 

given a gerbe on  with respect to , we can form a twist  of  as a group-scheme over , and consider the category . 

It will turn out that the above two operations are dual to each other with respect to the operation of 2-Fourier-Mukai transform. 

We will explore this relationship and its consequences for Langlands correspondence.

Laurent Lafforgue (Huawei)

Geometry and logic of subtoposes

After reviewing the multiple roles of toposes – as generalized topological spaces, as universal invariants, as pastiches of the category of sets and as incarnations of the semantics of first-order theories -, we shall recall the definition of the notion of subtopos and its double expression in terms of Grothendieck topologies and in terms of first-order logic. We shall stress the consequence of this double expression for translating first-order provability problems into problems of generation of Grothendieck topologies, and we shall introduce the natural geometric inner and outer operations on subtoposes.

After introducting these themes, we shall give a new presentation – based on some very general abstract nonsense – of the duality of Grothendieck topologies and subtoposes, and of the duality of topologies and closedness properties of subpresheaves.

Then we shall present two different general formulas expressing the Grothendieck topology generated by any given family of sieves or of covering families of morphisms. We shall also make more precise the constructive processes which allow to translate provability problems into topology generation problems.

Lastly, we shall study the inner geometric operations on subtoposes – union, intersection, difference – and the outer adjoint operations of push-forward and pull-back by topos morphisms. We shall prove that pull-back operations always respect not only arbitrary intersections but also finite unions of subtoposes, and that pull-backs by “locally connected” morphisms even respect arbitrary unions of subtoposes.

Part of the material of the lecture course is classical – borrowed from SGA4, from O. Caramello’s book “Theories, Sites, Toposes” and from other references – and part is new, borrowed from a joint paper coauthored with O. C. to appear soon under the title “Engendrement de topologies, démontrabilité et opérations sur les sous-topos”.

Michael Robinson (American University)

Practical systems modeling in categories using sheaves

Modeling practical systems using category theory (and especially topoi) can be daunting! There is plenty of expressive power, but in a sense it is too much. Fortunately, most practical systems also have a notion of topology, which provides strong modeling constraints. The theory of sheaves forms a functorial bridge between the topology of a model and its associated data. Famously, categories of sheaves are topoi. Therefore, although the job of modeling with sheaves is easier than purely with categories, it does not come at the expense of expressivity. Over the past decade or so, the topological data modeling community has developed tools that allow one to easily and effectively build sheaf models for common system models. This summer school session will explain how these work, both theoretically and practically.

It is also important to note that practical systems also have to deal with noise, errors, and uncertainty. Fortunately, sheaves are topological in their organization and in their representation of data. With a little forethought, sheaves can be used to handle the vagaries of practical systems. Moreover, in many scientific and engineering settings, there is geometric information. This allows one to measure how closely aligned the model posited by a sheaf and experimentally collected data are. The result is a statistical interpretation of data, models, and topology that has a practical algorithmic implementation… and this too is functorial! This summer school session will include software demonstrations to introduce participants to the use of these tools.

Isar Stubbe (Université du Littoral-Côte d’Opale)

Quantaloid-enriched categories for sheaf theory

A sheaf F on a locale L is commonly defined as a contravariant Set-valued functor on L that satisfies the gluing condition. Together with natural transformations, these sheaves form the objects and morphisms of the localic topos Sh(L). While every localic topos is a Grothendieck topos, the converse does not hold—there exist Grothendieck toposes that are not localic.

For any two elements (or sections) x and y in such a sheaf F (meaning that x is in Fu, and y is in Fv, for some u,v in L), we can measure the extent to which x equals y by computing the supremum of all w below both u and v on which the restrictions of x and y are equal (in the set Fw). This L-valued map on pairs of elements of F plays the role of a characteristic function for equality. In fact, the notion of sheaf can be reformulated in terms of such an L-valued map; and (together with an appropriate notion of morphism) these ‘L-sets’ form a category equivalent to Sh(L).

This latter formulation puts sheaf theory in the realm of many-valued logic, or more specifically, of quantaloid-enriched categories – and this will be the central theme of our lectures. Concretely, we shall first define quantales and quantaloids, explore some key universal constructions on these, and discuss several examples (noting in particular that locales are precisely ‘cartesian’ quantales). Then we will explain the fundamentals of quantaloid-enriched category theory, and show its flexibility and applicability through various examples. Finally we shall indicate how every Grothendieck topos Sh(C,J) is equivalent to a category of ‘Q-sets’ for an appropriate quantale Q, thereby showing that “every Grothendieck topos is quantalic”.

Invited speakers

Joseph Bernstein (Tel Aviv University)

Groups, Groupoids, Stacks and Representation Theory

In my talk I would like to introduce a new approach to Representation Theory.
Let G be an abstract group and k some field. A representation of the group G over the field k is usually defined as a pair (π, V), where V is a vector space over k and π is a morphism from G to Aut(V).
One of basic problems in Representation Theory is the study of the category Rep(G) of such representations.
In my talk I will explain that there is another natural way to describe this category.
Namely, the category Rep(G) is naturally equivalent to the category of sheaves Sh(B(G)) on some “geometric” object – the basic groupoid BG of the group G.
Thus we have two equivalent definitions of representations – standard one and categorical definition in terms of groupoids. I will explain that more sophisticated categorical description is more “correct” one. For example, it gives a more adequate description of the category Rep(G) in cases when we have some external symmetries.
The gap between these two definitions becomes much more profound when we move from the category of sets to other categories (more precisely – sites).
In this case the role of groupoids are played by stacks. So I propose to define the category of representations of a group G as the category of sheaves on the basic stack BG.
I will discuss how these things play out in the important case when G is an algebraic group over a local field.
My talk will partially follow my paper in arXiv:1410.0435. Relation between groups and groupoids is discussed in a paper by R. Brown “From Groups to Groupoids: a Brief Survey”, Bull. London Math. Soc. 19 (1987) 113-134.

Paolo Giordano (University of Vienna)

The Grothendieck topos of generalized smooth functions

The need to describe abrupt changes or response of nonlinear systems to impulsive stimuli is ubiquitous in applications. Also within mathematics, L. Hörmander stated: “In differential calculus one encounters immediately the unpleasant fact that not every function is differentiable. The purpose of distribution theory is to remedy this flaw; indeed, the space of distributions is essentially the smallest extension of the space of continuous functions where differentiability is always well defined”. We first describe the universal property of the space of distributions, but then we underscore the main deficiencies of this theory: we cannot evaluate a distribution at a point, we cannot make non-linear operations, let alone composition, we do not have a good integration theory, etc. We then present generalized smooth functions (GSF) theory, a nonlinear theory of generalized functions (GF) as used by physicists and engineers, where GF are ordinary set-theoretical maps defined on and taking values in a non-Archimedean ring extending the real field (this problem has been faced e.g. by: Schwartz, Lojasiewicz, Laugwitz, Schmieden, Egorov, Robinson, Colombeau, Rosinger, Levi-Civita, Keisler, Connes, etc.); GSF are closed with respect to composition so that nonlinear operations are possible; these operations coincide with the usual ones for smooth functions; all classical theorems of differential and integral calculus hold; we have several types of sheaf properties, and GSF indeed form a Grothendieck topos; we have a full theory of ODE, and general existence theorems for nonlinear singular PDE, e.g. the Picard-Lindelöf theorem for PDE; every Cauchy problem with a smooth PDE is Hadamard well-posed; we can generalize the classical Fourier method also to non-tempered GF (this problem has been faced e.g. by Gelfand, Sobolev); we have several applications in the calculus of variation with singular Lagrangians, elastoplasticity, general relativity, quantum mechanics, singular optics, impact mechanics (this problem has been faced by J. Marsden). We close by presenting a project in collaboration with several Japanese universities about how to apply GSF theory to have GF in diffeological spaces and hence to make homotopy theory where continuous functions can be treated as smooth functions, or to try to replicate synthetic differential geometry in the Grothendieck topos using nilpotent infinitesimals for this type of GF.

Peter Haine (University of California, Berkeley)

Reconstructing schemes from their étale topoi

In Grothendieck’s 1983 letter to Faltings that initiated the study of anabelian geometry, he conjectured that a large class of schemes can be reconstructed from their étale topoi. In this talk, I’ll discuss joint work with Magnus Carlson and Sebastian Wolf, generalizing work of Voevodsky, that proves Grothendieck’s conjecture. Specifically, we show that over a finitely generated field k of characteristic 0, seminormal finite type k-schemes can be reconstructed from their étale topoi. Over a finitely generated field k of positive characteristic and transcendence degree ≥ 1, we show that perfections of finite type k-schemes can be reconstructed from their étale topoi. Combined with joint work with Barwick and Glasman, and work of Makkai and Lurie on strong conceptual completeness, we deduce that such schemes can also be reconstructed from two different condensed categories of points of their étale topoi. Our talk will focus on the topos-theoretic aspects of these results.

Matthias Hutzler (University of Gothenburg)

Projective Space and Line Bundles in Synthetic Algebraic Geometry

Synthetic algebraic geometry is the study of schemes using an internal language of the Zariski topos. More precisely, Homotopy Type Theory (HoTT) is interpreted in a higher-topos variant of the Zariski topos in order to have a powerful language that can talk about higher homotopical objects just as easily as set-level objects. From this internal perspective, schemes such as for example projective space appear simply as certain h-sets without any added structure.

In the talk we present a synthetic version of the classical classification result for line bundles on projective space. The language of HoTT allows us to give a stronger variant of the classical statement, describing the 1-type of line bundles instead of its set-truncation, the Picard group. This is used to give a proof that requires nontrivial algebraic arguments only for the case of the projective line, and derives the general case by an interpolation argument.

Asgar Jamneshan (Koç University)

Some Applications of Toposes of Measure-Theoretic Sheaves

We construct toposes of sheaves on measure spaces and highlight the usefulness of interpreting certain structures from classical measure theory and functional analysis, combined with a Boolean internal logic, in applications to ergodic structure theory and vector duality.

Maxim Kontsevich (IHES)

What is the spectrum of quantum algebra?

For quantum algebras, like e.g. algebras of polynomial differential operators or q-difference operators, the “minimal” non-trivial modules are holonomic ones. These objects are not point-like, and correspond roughly to vector bundles on Lagrangian subvarieties in the semi-classical limit.

I’ll talk about various approaches to supports: 1) via behavior at infinity, 2) via reduction modulo large primes, 3) via multiplicative semi-norms in the non-archimedean case.

Mariano Hugo Luiz (University of São Paulo)

From quantales to a Grothendieck monoidal topology: Towards a closed monoidal generalization of topos

In this talk, we will present some recent developments associated with some PhD theses in IME-USP (Institute of Mathematics and Statistics, University of São Paulo, Brazil) on categories of sheaves over quantales and categories of quantale valued sets, returning to a theme of studies involving logic and categories carried out at IME-USP in the latter half of the 1990, but now from a new perspective: considering semicartesian and commutative quantales, as non-idempotent generalizations of locales (= complete Heyting algebras).

We will list some properties of the (monoidal) categories obtained, indicating some similarities and differences with the Grothendieck topos. The main goal of these efforts is to develop a closed monoidal but not cartesian closed generalization of the notion of elementary topos, in order to cover some mathematical situations (including generalizations of metric spaces), to enable an axiomatic study of these categories, and a general definition of their internal logic, which shows clues of being some form of linear logic.

A future goal is to establish a precise relationship between the present approach and the enriched category approach to sheaves over quantales (and quantaloids) developed by I. Stubbe.

Axel Osmond (Grothendieck Institute)

Morphisms and comorphisms of sites: double-categorical and profunctorial aspects

Geometric morphisms can be induced either from morphisms or comorphisms of sites, respectively in a contravariant and in a covariant way; the first are characterized through a cover-preservation property, the second through a cover-reflection property. As both define a relevant notion of 1-cells between sites, one may ask two questions:

• is there a proper way to mix them altogether in a single categorical structure on sites, and if so, does it help understanding the reason for which we have those twin classes of functors rather than a single one?
• is it possible to subsume them into a single notion jointly generalizing the cover-preservation and cover-reflection into a single condition?

In this talk, based on an ongoing work with Olivia Caramello, we will try to address those two questions.

In a first part, we will explain how morphisms and comorphisms, though they do not compose with each other, can be arranged as the horizontal and vertical 1-cells of a double-category of sites, and how the sheaf-topos construction defines a double-functor to the quintet double-category of topoi. We will also discuss some companion and conjoint constructions in this setting, as well as a link between this double-category of sites and the double-category of co-algebras, lax and colax morphisms for a 2-comonad.

In a second part, we will try to answer to the second question through notions of continuity for distributors (a.k.a. profunctors); building on Bénabou theory of flat distributors and refining a previous definition of Johnstone and Wraith, we prove an equivalence between continuous distributors between sites and geometric morphisms between the corresponding sheaf topoi, and discuss how this notion relates respectively to morphisms and comorphisms of sites.

Hans Riess (Duke University)

Towards Categorical Diffusion

John Baez and others have noted a compelling analogy between adjoint functors in category theory and adjoints of linear operators in Hilbert spaces. This analogy is particularly striking when considering enriched adjunctions, where the hom mimics an inner product. On the other hand, Hodge theory, which bridges PDEs with geometry and topology, fundamentally relies on the existence of an adjoint operator within the de Rham complex. In this talk, we explore ongoing efforts to categorify the Hodge Laplacian, focusing on the connection Laplacian—a special case involving parallel transport between tangent spaces of a manifold. We propose a framework for parallel transport and a connection Laplacian within the setting of presheaves on preorder into the category of V-enriched categories. Finally, we outline preliminary examples of this theory, with potential applications to logic, formal concept analysis, and tropical geometry.

Michael Shulman (University of San Diego)

Internal languages of diagrams of toposes

Internal languages are a powerful tool for studying toposes, but traditionally they can only be applied to a single topos at a time, whereas frequently we are interested in a number of toposes related by a diagram of geometric morphisms. The collection of internal languages of the toposes in such a diagram, together with syntactic operations relating them corresponding to the direct and inverse image functors of the geometric morphisms and transformations, form a “modal” type theory, with the functor operations known as “modalities”. The first general modal type theories, applicable to a diagram of arbitrary shape, have recently been formulated by Gratzer, Kavvos, Nuyts, and Birkedal, but interpreting these theories in a diagram of toposes appears to require that all the geometric morphisms be essential. I will show how this requirement can be avoided, by generalizing the construction of a “fibration of sites” presenting a single geometric morphism to a presentation of an arbitrary diagram of such.

Contributed talks

Thiago Alexandre (University of São Paulo and IMJ, Paris)