# LHC days 2023

The fifth edition of the *LHC days* will take place on **Tuesday 6 and Wednesday
7 June 2023** at IRIF. They will be followed by a conference in honor of
François Métayer, which should also be of
interest.

# Important information

A national strike is planned on the 6 June. We will make our possible to adapt the planning accordingly, in particular if it has a large impact on transportation.

# Location

The conference will take place in the amphitheater 4C of *halle aux farines*:

# Invited speakers

- Bérénice Delcroix-Oger:
*Poset homology and operads* - Paul-André Melliès:
*The rabbit calculus: convolution products on double categories and categorification of rule algebra*

# (Provisional) program

**Tuesday 6 June 2023**

We have voluntarily reduced most of the program on Tuesday in order to help people accommodate with the planned strikes which are likely to affect transportation much, taking also into account a large expression of support for the ongoing social movement in our community.

16:45 *Welcome*

17:00 Guodong Zhou

17:30 Luidnel Maignan

18:00 Bryce Clarke

18:30 *End of the day (and beginning of the night)*

We will try to organize a dinner in a restaurant for those who want (at participant’s cost).

**Wenesday 7 June 2023**

9:00 Thibaut Benjamin

9:30 Hugo Herbelin

10:00 Eric Finster

10:30 *Break*

11:00 Bérénice Delcroix-Oger (invited talk)

12:00 Lutz Strassburger

12:30 *Lunch*

14:00 Paul-André Melliès (invited talk)

15:00 Vincent Moreau

15:30 *Break*

16:00 Thomas Ehrhard

16:30 Clemens Berger

17:00 El Mehdi Cherradi

17:30 *End of the (long) day*

# Abstracts

**Thibaut Benjamin: A Coherator For Pre-Cubical Weak ω-Categories** / slides

We present a definition of cubical weak ω-categories, that relies on pre-cubical sets. This definition is inspired by the Grothendieck-Maltsiniotis definition of globular weak ω-categories. In particular, our underlying shapes do not have degeneracies, symmetries, or connections built-in, instead, we retrieve those by requiring additional structure. Following Grothendieck’s insight, we define cubical weak ω-categories as presheaves preserving certain limits over a well-chosen category called a coherator.

**Clemens Berger: On the profinite fundamental group of a connected Grothendieck topos.**

We show that for a convenient notion of finiteness, finitely generated, connected Grothendieck toposes are equivalent to classfiying toposes of profinite groups.

**El Mehdi Cherradi: The smothering model structure and prederivators as (∞,1)-categories**
/ slides

The notion of smothering functor introduced by Riehl and Verity defines a class
of functors which extends the usual notion of equivalence of categories by
relaxing the faithfulness condition. These functors appear naturally as
comparison functors between homotopy categories, for instance the comparison
functor relating the homotopy category of a pullback of quasicategories to the
pullback of the homotopy categories of these quasicategories. In this work, we
consider a slight strengthening of the original definition of smothering
functors, which we refer to as stably smothering, that actually form the trivial
fibrations of a right Bousfield localization of the natural model structure on
Cat. Starting from the smothering model structure on Cat and the induced Reedy
model structure on the category of functors *∆ᵒᵖ → Cat*, we construct a model
structure on the category prederivators and a Quillen equivalence with the Joyal
model structure on simplicial sets.

**Bryce Clarke: Lifting and lenses** / slides

Delta lenses are functors equipped with a functorial choice of lifts, and generalise the notion of a split opfibration. It is well-known that split opfibrations are algebras for a monad which arises from an algebraic weak factorisation system (awfs) on Cat, and that left-adjoint-right-inverse functors lift against split opfibrations. In this talk, I will show that delta lenses are also algebras for a monad arising from an awfs on Cat, and demonstrate the close relationship with both the awfs for split opfibrations and the comprehensive factorisation system. In particular, I will construct the free delta lens on a functor, and characterise the additional structure on a functor required for it to lift against a delta lens. This talk is based on the recent preprint arXiv:2305.02732.

**Bérénice Delcroix-Oger: Poset homology and operads** / slides

To any poset, or partially ordered set, can be associated a simplicial set called its order complex, which enables one to define the homology of the poset as the homology of its associated order complex. The key example developed in this talk is the (set) partition poset. A. Joyal pointed out in the 1980s that the character of the action of the symmetric group on the homology of the partition poset, computed by Stanley and Hanlon at the beginning of the 1980s, is the same as the character for Lie operad, which encodes Lie algebras. B. Fresse identified the chain complex of the partition poset with the bar construction for Comm operad. B. Vallette extended this construction to decorated partition poset and draw a link between an algebraic property on operads and a topological property on posets.

In this talk, we will first introduce all the needed notions on posets, species and operads. We will then explain the longterm relationship between partition posets and Lie operad before explaining the newly discovered relationship between hypertree poset and Post-Lie operad. It is based on a joint work with C. Dupont.

**Thomas Ehrhard: Coherent differentiation**
/ slides

The denotational models which account for the determinism of computation feature a partial addition of morphisms: it does not make sense to add “true” and “false” in the type of booleans. This suggests that differentiation, as axiomatized for instance in differential linear logic, is incompatible with determinism since the differential calculus uses addition in a crucial way (Leibniz rule, Taylor formula etc). I will present Coherent Differentiation which seems to solve this contradiction, allowing to define differentiation in categories such as (probabilistic) coherence spaces.

**Eric Finster: The Baez-Dolan +-construction for Generalized Algebraic Theories**

**Hugo Herbelin: A parametricity-based construction of semi-simplicial and semi-cubical sets**
/ slides

On one side, presheaves on a direct category have the ability to be defined in either a “fibered” way (that is a functor) or an “indexed” way (that is as a stream of families denoting the presheaf seen as a projective limit, where each family of the stream depends on the previous ones). On another side, both the semi-simplicial and semi-cubical index categories can be seen as particular cases of a more general notion of “freely-parametric” index category of arity respectively 1 (unary parametricity, similar to realisability) and 2 (binary parametricity, as in Reynolds’ original approach). We give a uniform indexed construction of both semi-simplicial and semi-cubical sets reflecting the way Reynolds’ parametricity of arity 1 and 2 is defined. The construction is a step towards building parametricity-based models of cubical type theory. It is incidentally fully formalised in Coq.

**Luidnel Maignan: Non Determinism in Spatialized Systems, Generically**

In my works with Antoine Spicher and Alexandre Fernandez, we consider an abstraction of space where we only think of it through the way it limits the knowledge you can have about the state of a system. This is formalized through a category of object representing something similar to a knowledge state about the system. In a recent MFCS paper, we showed that non-determinism is, in some categorical sense, a particular case of the general definition. We did it on the concrete example of non-deterministic Lindenmeyer systems, and showed how the 2-monad of families is necessary to express this abstraction. For this presentation, I would also like to quickly explain how this is related to different categorical constructions: presheaves, profunctors, and left fibrational spans.

**Paul-André Melliès: The rabbit calculus: convolution products on double categories and categorification of rule algebra**
/ slides

Reporting on recent joint work with Nicolas Behr and Noam Zeilberger, I will describe the rabbit calculus, a convolution product over preasheaves of double categories motivated by compositional categorical rewriting theory. As I will explain, the convolution product generalizes to every double category the usual Day tensor product of presheaves of monoidal categories. One interesting aspect of the construction is that this convolution product is in general only oplax associative. For that reason, several classes of double categories will be identified for which the convolution product is not just oplax associative, but fully associative. This includes in particular framed bicategories on the one hand, and double categories of compositional rewriting theories on the other. For the latter, we establish a formula which justifies the view that the convolution product categorifies the rule algebra product.

**Vincent Moreau: Profinite λ-terms and parametricity**
/ slides

The aim of this work is to combine profinite methods and models of the λ-calculus to obtain a notion of profinite λ-term which, we show, lives in perfect harmony with the principles of Reynolds parametricity.

**Lutz Strassburger: The Surprising Mystery of Pomset Logic and BV** / preprint

There is no abstract because otherwise it wouldn’t be surprising, and it wouldn’t be a mystery.

**Guodong Zhou: The homotopy theory of operated algebras** / slides

The talk is a survey of our recent results on the homotopy theory of operated
algebras such as Rota-Baxter associative (or Lie) algebras and differential
associative (or Lie) algebras. We make explicit the Kozul dual homotopy
cooperads and the minimal models of the operads governing these operated
algebras. As a consequence the *L∞* structures on the deformation complexes are
described.

# Participants

- Matteo Acclavio (University of Southern Denmark)
- Quentin Aristote (IRIF)
- Nicolas Behr (CNRS, IRIF, Université Paris Cité)
- Thibaut Benjamin (University of Cambridge)
- Arij Benkhadra (Université de Paris Nanterre)
- Clemens Berger (Université Côte d’Azur)
- Hugo Cadière (Université Jean Moulin lyon 3)
- Cameron Calk (Laboratoire d’Informatique et Systèmes (Univ. Aix-Marseille))
- El Mehdi Cherradi (IRIF)
- Younggi Choi (Seoul National University, Department of Mathematics Education)
- Jules Chouquet (LIFO (Université d’Orléans))
- Bryce Clarke (Inria Saclay)
- Sophie d’Espalungue (Paris Cité)
- Alexander De Klerck (KU Leuven)
- Marc de Visme (Loria, Nancy)
- Bérénice Delcroix-Oger (IMAG, Université de Montpellier)
- Raffaele Di Donna (IRIF, Université Paris Cité, Università Roma Tre)
- Sylvain Douteau (IRIF)
- Aloÿs Dufour (Université Paris-Nord)
- Thomas Ehrhard (IRIF)
- Raül Espejo Boix (Université de Rennes 1 et IRISA)
- Uli Fahrenberg (LRE, EPITA)
- Eric Finster (University of Birmingham)
- Laura Fontanella (Université Paris Est Créteil)
- Zeinab Galal (LIP6, Sorbonne Université)
- Guillaume Geoffroy (Université Paris Cité - IRIF)
- Pierre Giraud (équipe Gallinette, INRIA (centre de Rennes –Loire atlantique) (hébergé à Nantes Université))
- Adrien Guatto (IRIF)
- Amar Hadzihasanovic (Tallinn University of Technology)
- Hélène Han (IRIF (en stage de M1))
- Elies Harington (Ecole Polytechnique)
- Grant Harvey (Queen’s University Belfast)
- Hugo Herbelin (IRIF - INRIA)
- Moana Jubert (IRIF, Inria, CNRS, Université Paris-Cité)
- Stefano Kasangian (Università di Milano)
- Roman Kniazev (LIX)
- Praphulla Koushik (IISER Pune)
- Adrienne Lancelot (INRIA & Lix)
- Samuel Lavenir (EPFL)
- Louis Lemonnier (LMF, Université Paris-Saclay)
- Quan Long (ENS Paris Saclay)
- Luidnel Maignan (LMF/ENS Paris-Saclay & LACL/Univ. Paris-Est Créteil)
- Georges Maltsiniotis (IMJ-PRG)
- Aníbal Medina (Paris 13)
- Thomas Jan Mikhail (Autonomous University of Barcelona)
- Mariana Milicich (UPC)
- Samuel Mimram (École polytechnique)
- Marianela Morales (INRIA Saclay & École Polytechnique)
- Vincent Moreau (IRIF, Université Paris Cité, Inria Paris)
- François Métayer (IRIF)
- Lê Thành Dũng (Tito) Nguyễn (École normale supérieure de Lyon)
- Guglielmo Nocera (Université Paris 13)
- Emile Oleon (LIX)
- Hugo Paquet (LIPN)
- Stiéphen Pradal (University of Nottingham)
- Francesca Pratali (Université Sorbonne Paris Nord)
- Manuel Rivera (Purdue University)
- Gabriel Saadia (Stockholm University)
- Elena Sendroiu (Independent researcher)
- Lutz Strassburger (Inria Saclay)
- Alexis Terrassin (Université Paris Cité)
- Kai Wang (East China Normal University)
- Vladimir Zamdzhiev (Inria)
- Noam Zeilberger (LIX)
- Guodong Zhou (East China Normal University)

# Contacts

In case of any question you can contact the organizers Pierre-Louis Curien (local organizer), Samuel Mimram and Lionel Vaux.