Algebra and computation (27 February – 2 March)

Talk list

Lectures

  • Vladimir Dotsenko (Université du Luxembourg, Luxembourg), Rewriting methods for associative algebras and operads.

    I. Associative algebras and Groebner bases. Bergman's Diamond lemma, and an algorithm for constructing Groebner bases that it implies. (Digression: the case of commutative algebras, and systems of polynomial equations.)

    II. Symmetric and nonsymmetric operads. Examples. Generalisation of Groebner bases for the case of nonsymmetric operad. Trees and word sequences. Failure of rewriting methods for symmetric operads.

    III. Shuffle operads. Monoidality of the forgetful functor, and its consequences for the symmetric case. Admissible ordering of shuffle trees. Examples of Groebner bases. Koszul duality and Hoffbeck's PBW criterion.

    IV. Groebner bases and free resolutions. Module resolutions: the theorem of Anick--Groves--Squier. Algebra resolutions. A-infinity algebras.

    • Tuesday 28, 9:00
    • Wednesday 29, 11:00
    • Friday 2, 9:00
  • Timothy Porter (University of Wales, Bangor, UK), Rewriting and Homotopy.

    I. & II. Some combinatorial group theory and low dimensional homotopy: Presentations, identities among relations, crossed modules, and crossed resolutions. Homological and homotopical syzygies. Higher generation by subgroups (Abels and Holz). Examples.

    III. An introduction to homotopy coherence and its rewriting aspect. Homotopy coherence and the resolution of a category. Examples of homotopy coherent diagrams and the homotopy coherent nerve. Quasi-categories. The link with rewriting.

    IV. How to adapt away from the group theory case... brief discussion of directed homotopy, polygraphs etc. and how to work with rewriting and syzygies in the non-group case, some pointers to combinatorial category theory.

    • Monday 27, 9:00
    • Tuesday 28, 11:00
    • Thursday 1, 9:00
    • Friday 2, 11:00
  • Michael Warren (IAS, Princeton, USA), Homotopy and type theory.

    In this series of lectures I will introduce some of the fundamental concepts and results arising from recent research relating homotopy theory, higher-dimensional category theory and type theory. Among the topics to be covered are the following: The homotopy theoretic interpretation of type theory, the Univalence Axiom and its consequences, Voevodsky's model of Univalence, and the generation of homotopy theoretic structures from type theory.

    • Monday 27, 11:00
    • Wednesday 29, 9:00
    • Thursday 1, 11:00

Invited talks

  • Dimitri Ara (Université Paris 7), On Grothendieck infinity-groupoids.

    In Pursuing Stacks, Grothendieck defines a notion of (weak) ∞-groupoid and constructs a fundamental ∞-groupoid functor that associates to every topological space its fundamental ∞-groupoid. He conjectures that this functor induces an equivalence on the homotopy categories, and in particular that ∞-groupoids classify homotopy types. In this talk, we will explain Grothendieck's definition and we will give a precise statement of Grothendieck's conjecture. If time allows, we will study the homotopy theory of Grothendieck ∞-groupoids.

    • Thursday 1, 17:30
  • Albert Burroni (Université Paris 7), Analyse de la catégorie des polygraphes.

    Les polygraphes ont été introduit par Street sous le nom de computads. De manière indépendante, à la fin des années 80, j'ai introduit cette notion pour donner un sens précis à un théorème : toute théorie de Lawvere (finitaire) est définissable comme un 2-monoïde (catégorie monoïdale stricte) et, surtout, cette théorie de Lawvere est de présentation finie si et seulement si le 2-monoïde correspondant est de présentation finie (cette présentation, étant un 3-polygraphe).

    Pour tout entier n, les n-graphes, les n-multigraphes, les n-polygraphes forment une suite naturelle de structures croissantes en complexité. Mon but est de les comparer, d'analyser celle de polygraphe (notamment le problème du pasting) et de montrer les difficultés de calculs rencontrées avec ce type de structures qui sont partout présentes dans toutes les machines ou processus de calcul de l'informatique théorique, mais souvent de manière cachée.

    • Tuesday 28, 14:00
  • Marcelo Fiore (University of Cambridge, UK), Second-Order Algebra and Generalised Polynomial Functors.

    The theme of this work is algebraic structure in settings richer than that of universal algebra. In this talk, I will consider an example and foundations. Starting with the example, I will introduce Second-Order Algebra: the algebra of languages with variable binding. This I will study from the viewpoints of universal algebra (equational presentations and algebraic models), equational logic (second-order deduction and rewriting), and categorical algebra (algebraic theories and functorial semantics). In this context, as well as in more general algebraic settings for languages with polymorphism and dependency, generalised polynomial functors between presheaf categories arise as foundational structures. I will present their basic theory, including differentiation.


    References

    • Monday 27, 15:00
  • Nicola Gambino (University of Palermo), Homotopy-initial W-types.

    It is well-known that within the extensional version of Martin-Lof type theory the deduction rules for W-types are equivalent to the existence of initial algebras for polynomial functors. The aim of the talk is to examine what happens to this characterization in the intensional version of Martin-Lof type theory. I will also show how some inductive types can be represented as W-types assuming a propositional version of the principle of function extensionality. The talk is based on joint work with Steve Awodey and Kristina Sojakova.

    • Thursday 1, 14:00
  • Stéphane Gaussent (Université Nancy 1), Coherent presentations and actions on categories.

    I will report on a joint work with Yves Guiraud and Philippe Malbos. For a given presentation of a monoid M, we consider its associated 2-polygraph. Then using a machinery based on higher rewriting theory, we obtain a homotopy basis. This homotopy basis is exactly the piece of information one needs to get the coherence diagrams involved in a (clever) definition of an action of M on a category. In some good cases, it is possible to simplify this basis. Namely, if we start with the Deligne's presentation of the positive braid monoid, we obtain a simplified version of a result of Deligne.

    • Tuesday 28, 16:30
  • André Joyal (Université du Québec à Montréal), On the enrichement of operads over cooperads.

    We show that the category of operads is enriched over the category of cooperads, and that the latter is symmetric monoidal closed. If time permits, we will discuss applications to the bar-cobar duality (joint work with Matthieu Anel).

    • Monday 27, 14:00
  • Yves Lafont (Université Aix-Marseille 2), Diagrammatic syntax for algebra.

    In algebra, rewriting is a model of symbolic computation: starting from a presentation by generators and relations, one tries to build a convergent rewrite system in order to compute normal forms. In proof theory, rewriting is a model of cut elimination, which corresponds to an execution of a proof (seen as a program). Those two approaches seem to be quite different, but there are similarities between them:

    • In both cases, people know use diagrams (or nets) rather than words or trees.
    • In both cases, people know use linear combinations (or formal sums) of such diagrams, which I call Σ-diagrams.

    I will give an overview of rewriting techniques in those new frameworks.

    • Monday 27, 16:30
  • Jean-Louis Loday (CNRS and Zinbiel Institute of Mathematics, Strasbourg), Homotopy Transfer Theorem.

    Let (A,d) be a chain complex and (V,d) be a sub-complex that we assume to be a deformation retract of A. If A bears some algebraic structure (like module over an associative algebra, or an algebra structure of some type), then it is not always the case that V is endowed with this very same structure. However, in good cases, there is a notion of ''structure up to homotopy'' which is supported by V. The game consists in unraveling this ''structure up to homotopy'' when given a specific algebraic structure on A. For instance if (A,d) is an associative differential graded algebra, then (V,d) is an A-infinity algebra. We investigate a large class of examples and we show how the methods of rewriting systems help to solve this problem. We explain the relationship with spectral sequences, Koszul duality, Grobner-Shirshov basis, crossed modules and even DNA.

    • Wednesday 29, 14:00
    • Friday 2, 14:00
  • Paul-André Melliès (CNRS, Université Paris 7), Braided notions of dialogue categories .

    A dialogue category is a monoidal category equipped with an exponentiating object bot called its tensorial pole. In a dialogue category, every object x is thus equipped with a left negation x --o bot and a right negation bot o-- x. An important point of the definition is that the object x is not required to coincide with its double negation. In this talk, I will define three non-commutative notions of dialogue categories -- called helical, cyclic and balanced dialogue categories. I will then explain how to see the category of left H-modules of arbitrary dimension on a ribbon Hopf algebra H as a balanced dialogue category Mod(H) whose tensorial pole bot is provided by the underlying field k. I will also explain how to recover from this basic observation the well-known fact that the full subcategory of finite dimensional left H-modules defines a ribbon category.

    • Thursday 1, 16:30
  • Samuel Mimram (CEA Saclay), A polygraphic presentation of first-order causality.

    Game semantics aims at describing the interactive behaviour of proofs (for instance in first-order logic) by interpreting formulas as games on which proofs induce strategies. One of the main difficulties that has to be faced when constructing such semantics is to make them precise by characterizing definable strategies - that is strategies which actually behave like a proof. This characterization is usually done by restricting to the model to strategies satisfying subtle combinatory conditions such as innocence, whose preservation under composition is difficult to show. Here, we present an original methodology to achieve this task which requires to combine tools from game semantics, rewriting theory and categorical algebra. We introduce a diagrammatic presentation of definable strategies by the means of generators and relations: those strategies can be generated from a finite set of ``atomic'' strategies and that the equality between strategies generated in such a way admits a finite axiomatization, which is a variation of bialgebra laws.

    • Tuesday 28, 15:00
  • Pierre Rannou (Université Aix-Marseille 2), Bialgebras and ∑-diagrams.

    It is well-known that any free (noncommutative) algebra has a canonical structure of (cocommutative) bialgebra. Using rewriting of Σ-diagrams by interaction rules, we prove a similar result for generalized bialgebras satisfying some conditions which are similar to the confluence of critical peaks.

    • Monday 27, 17:30
  • Bruno Vallette (Université de Nice-Sophia Antipolis), Associative algebras.

    Homotopy associative algebra

    When one wants to transfer associative algebra structures though homotopy equivalences, one automatically discovers the notion of homotopy associative algebra. In order to conceptually understand such a result, and later to generalize it, I will introduce a simple operad, based on planar rooted trees. Finally, I will explain how to build a resolution of this operad, due to Jim Stasheff, which is based on polytopes, called the associaedra.

    Koszul duality and Diamond Lemma

    Starting from the symmetric algebra as the toy model for quadratic algebras, I will introduce the notion of Poincaré-Birkhoff-Witt bases. I will then describe the Koszul duality theory for associative algebras. Finally, I will show that an algebra which admits a PBW basis is Koszul. This result is based the Diamond lemma and the rewriting method thereby drawing a concrete link with the other talks of the workshop.

    • Tuesday 28, 17:30 — Homotopy associative algebra
    • Thursday 1, 15:00 — Koszul duality and Diamond Lemma