Research
My research lies in the foundations of higher category theory. For nearly twenty years, I have been developing algebraic and combinatorial approaches to weak higher-dimensional structures, with particular emphasis on monads, higher operads, Grothendieck coherators, higher transformations, and the comparison between globular, cubical and multiple geometries.
A recurring theme of my work is that higher categories should not only be described through their objects and compositions, but through an entire hierarchy of higher morphisms: weak ∞-functors, weak natural ∞-transformations, modifications, and transformations of arbitrarily high dimension. This naturally leads to a more ambitious question: can weak ∞-categories themselves be organized into a weak ∞-category?
1-- Globular higher categories and higher transformations
My first work in higher category theory was developed in the globular setting.
I constructed algebraic models of globular weak ∞-functors and weak higher transformations using monads, and subsequently developed operadic descriptions of these structures. This led to a coglobular sequence of globular operads
whose algebras describe successive dimensions of higher categorical structure: globular weak ∞-categories, globular weak ∞-functors, globular weak natural ∞-transformations, and higher transformations.
This hierarchy suggested a fundamental recursive problem. Ordinary categories, functors and natural transformations organize themselves into higher categorical structures. One may therefore ask whether the same phenomenon continues in all dimensions: does the totality of globular weak ∞-categories itself carry the structure of a globular weak ∞-category?
I reduced this question to a precise problem concerning a coendomorphism operad associated with the coglobular sequence above. In particular, the contractibility of this coendomorphism operad would provide an action of the operad of globular weak ∞-categories and hence produce the desired higher categorical structure.
This problem has remained one of the conceptual threads running through my research.
2-- Monads, operads and Grothendieck coherators
An important aspect of my work is the use of several algebraic languages for describing the same higher-dimensional phenomena.
I have studied higher categorical structures through Categorical Stretchings of Penon, Batanin operads and Grothendieck coherators. These approaches emphasize different aspects of the theory: monads provide explicit algebraic descriptions, operads encode higher compositions and their coherences, while Grothendieck coherators describe weak higher structures through higher-dimensional pasting operations and universal lifting constructions.
This multiplicity of viewpoints is deliberate. One of my long-term objectives is to understand how these different algebraic formalisms interact and to determine which constructions are intrinsic to higher category theory rather than artifacts of a particular presentation.
In particular, I constructed algebraic models of weak (∞,m)-categories for arbitrary m, first through monadic methods and later through Grothendieck coherators. I have also investigated the corresponding hierarchy of higher transformations.
3-- From globular to cubical higher category theory
A second major direction of my research concerns cubical higher categories.
The motivation is both algebraic and homotopical. Globular geometry provides a particularly natural language for higher categorical composition, whereas cubical geometry possesses structures that are closely related to classical constructions in homotopy theory.
I therefore developed a cubical counterpart of the operadic theory of Batanin. This required constructing cubical analogues of the combinatorial objects underlying globular higher operads, including a cubical version of Θ₀ describing cubical pasting diagrams.
This led to algebraic and operadic models of cubical weak ∞-categories and cubical weak ∞-groupoids, as well as to cubical Grothendieck coherators.
An important technical part of this programme concerns the monads governing cubical strict ∞-categories. Establishing their cartesian properties makes it possible to develop a genuine theory of cubical higher operads and to compare operadic, monadic and coherator-based descriptions of cubical weak higher structures.
4-- Higher groupoids and homotopy types
The cubical theory also led naturally to homotopy theory.
I constructed a fundamental cubical weak ∞-groupoid functor from topological spaces to cubical weak ∞-groupoids. This construction raises the question of whether cubical weak ∞-groupoids provide an algebraic model for homotopy types.
More generally, I am interested in Grothendieck's homotopy hypothesis: the principle that homotopy types should be modeled by weak ∞-groupoids.
One strategy I have investigated consists in comparing different combinatorial geometries for higher groupoids. In particular, my work suggests a network of relationships
multiple ↔ cubical ↔ globular
through which homotopical information might be transported from one higher-categorical geometry to another.
This perspective is closely related to the search for model structures on categories of higher groupoids and to the possibility of comparing these structures through Quillen equivalences.
5-- Higher spans and related algebraic structures
Another part of my work concerns higher spans and the higher categorical structures naturally generated by them.
Spans are among the simplest constructions in which higher-dimensional compositions arise naturally. Their iterated versions provide important examples of higher categories and are closely connected with monoidal higher structures, higher monads and operadic constructions.
My work in this direction is related to the study of cartesian monads and to the interaction between multiple and cubical higher categories.
6-- Higher monads
A further direction concerns the theory of higher monads.
Classical monad theory already provides a powerful mechanism for encoding algebraic structures categorically. In higher dimensions, however, a monad should admit several levels of algebraic structure, involving strict, pseudo, lax and higher coherent notions of algebra.
I have been interested in understanding these possible notions of algebra for n-monads and in their potential applications to the classification of higher categorical structures and, ultimately, homotopy types.
This direction is connected with the formal theory of monads initiated by Bénabou and Street, with two-dimensional monad theory, and with the problem of extending these ideas systematically to arbitrary dimension.
7-- Current perspective
The different parts of this programme are connected by a common question:
How can higher-dimensional mathematical structures be described by explicit algebraic and combinatorial mechanisms while retaining their homotopical content?
My approach has been to move repeatedly between several levels:
a-- higher categories and higher transformations;
b-- monads and their algebras;
c-- higher operads and their actions;
d-- Grothendieck coherators and pasting diagrams;
e-- globular, cubical and multiple geometries;
f-- higher groupoids and homotopy types.
More recently, I have returned to the problem of higher transformations in the cubical setting. In particular, I am interested in developing a systematic theory of strict and weak cubical higher transformations and in understanding their inductive structure.
The long-term objective is a coherent algebraic architecture in which weak ∞-categories, weak ∞-functors, higher transformations and weak ∞-groupoids arise as different levels of a single recursive higher-dimensional structure.
Selected Publications and Mathematical Contributions
The following articles trace the development of my research programme, from the algebraic foundations of globular higher transformations to cubical weak higher categories, higher operads, and Grothendieck coherators.
1. Définition algébrique des cellules non strictes
Les Cahiers de Topologie et Géométrie Différentielle Catégoriques, 49(1) (2008), 1–68.
In this first article, written in French, I introduced monadic descriptions of globular weak higher transformations using variations of the categorical stretchings introduced by Jacques Penon in 1998. The objective was to provide an algebraic framework for higher-dimensional categorical cells and their transformations.
2. Operadic Definition of Non-Strict Cells
Les Cahiers de Topologie et Géométrie Différentielle Catégoriques, 52(4) (2011), 269–316.
In this article, I developed operadic descriptions of globular weak higher transformations, using Michael Batanin's theory of globular operads and the formalism of T-categories developed by Tom Leinster.
An important historical clarification is necessary. In 2012, André Joyal pointed out an obstruction in the contraction formalism used in this article: the coloured operads did not possess sufficiently rich contraction operations to encode all the expected coherences of globular weak higher transformations. We investigated this problem together without finding a satisfactory general solution at that time.
In 2015, I developed a corrected formalism, presented in subsequent work. The underlying collections constructed in the 2011 article remain valid, but the original contraction structure must be modified. See Article 6 below, Steps Toward the Weak Higher Category of Weak Higher Categories in the Globular Setting.
3. An Algebraic Definition of (∞, n)-Categories
Theory and Applications of Categories, 30(22) (2015), 775–807.
In this article, we constructed, for each non-negative integer n, a monad on the category of globular sets whose algebras provide models of globular weak (∞, n)-categories.
The case n = 0 gives an algebraic description of globular weak ∞-groupoids. We thus proposed a second type of algebraic models for globular weak ∞-groupoids after the coherator approach by Grothendieck.
Our construction provides the first monadic approach to globular weak ∞-groupoids and, more generally, to globular weak (∞, n)-categories, far from the usual technology of simplicial sets and homotopy devices, and also it is still not clear for us why the Grothendieck's approach could be seen as algebras for a monad.
4. ω-Operads of Coendomorphisms and Fractal ω-Operads for Higher Structures
Categories and General Algebraic Structures with Applications, 3(1) (2015), 65–88.
This article introduces the framework of fractal ω-operads, motivated by the idea that higher categorical structures may exhibit a form of self-similarity.
A central construction involves the coendomorphism ω-operad associated with a coglobular object of ω-operads. The objective is to understand how such constructions can encode higher transformations and provide a mechanism for organizing higher categories themselves into higher categorical structures.
5. Operads of Higher Transformations for Globular Sets and for Higher Magmas
Categories and General Algebraic Structures with Applications, 3(1) (2015), 89–111.
In this article, I developed examples of fractal ω-operads associated with higher transformations.
The work provides four examples intended to support the conjectural contractibility of a particular coendomorphism operad, referred to as the violet operad.
These examples are closely connected with the possibility of constructing a globular weak ∞-category of globular weak ∞-categories.
6. Steps Toward the Weak Higher Category of Weak Higher Categories in the Globular Setting
Categories and General Algebraic Structures with Applications, 4(1) (2016), 9–42.
This article provides the corrected operadic formalism for weakened globular strict higher transformations, addressing the difficulties identified in the contraction structures of my earlier work.
It forms part of my programme to establish the algebraic conditions under which globular weak ∞-categories, weak ∞-functors, higher natural transformations, modifications, and higher-dimensional transformations can themselves be assembled into a globular weak higher category.
In particular, the article identifies precise conditions under which certain globular operads possess the fractal property, thereby isolating important combinatorial difficulties in the construction of the globular weak ∞-category of globular weak ∞-categories.
7. Algebraic Models of Cubical Weak ∞-Categories with Connections
Categories and General Algebraic Structures with Applications, 16(1) (2022), 143–187.
This article develops algebraic models of cubical weak ∞-categories with connections, using a cubical analogue of Penon's categorical stretchings.
The central construction is based on cubical contractions: additional operations that weaken the strict composition laws of cubical strict ∞-categories. Unlike their globular counterparts, these contraction operations are defined inductively.
In dimension two, the construction recovers the weak double category studied by Dominic Verity in his thesis.
8. Algebraic Models of Cubical Weak Higher Structures
Categories and General Algebraic Structures with Applications, 16(1) (2022), 189–220.
This article extends the monadic approach to several types of cubical weak higher structures.
In particular, I construct algebraic models of cubical weak ∞-groupoids with connections as algebras for a monad on the category of cubical sets.
I also develop corresponding descriptions of cubical weak ∞-functors and cubical weak natural ∞-transformations. These are represented by algebras for monads acting on suitable finite products of the category of cubical sets.
The underlying cubical sets are considered without requiring degeneracy operators as part of their initial structure.
9. Combinatorial Approach to the Category Θ₀ of Cubical Pasting Diagrams
Categories and General Algebraic Structures with Applications, 21(1) (2024), 19–68.
This article develops the combinatorics of a small category Θ₀ whose objects represent cubical pasting diagrams, playing a role analogous to that of globular trees in globular higher category theory.
A suitable subcategory is used to describe the monad whose algebras are cubical sets equipped with connections. I prove that this monad is cartesian, providing a first important example of the use of our combinatorics.
10. The Coherator Θ_W^∞ of Cubical Weak ∞-Categories with Connections
Categories and General Algebraic Structures with Applications, 21(1) (2024), 69–126.
This article uses the combinatorics of cubical pasting diagrams to give an explicit description of the monad governing cubical strict ∞-categories with connections and to establish its cartesianity.
This result resolves a conjecture I formulated during my work at the Institut des Hautes Études Scientifiques in 2018 and provides a foundation for the construction of cubical higher operads.
The article also develops Grothendieck-style cubical coherators whose set-valued models describe cubical weak ∞-categories and cubical weak ∞-groupoids with connections.
I outline how the same approach could be extended to cubical weak ∞-functors and cubical weak natural ∞-transformations.
Finally, I propose a broader programme based on cocubical objects and coendomorphism operads. The central conjectural idea is that suitable actions of these operads could produce a self-similar higher-dimensional structure: a cubical weak ∞-category of cubical weak ∞-categories.
Such a construction could eventually contribute to algebraic approaches to cubical weak ∞-toposes and cubical weak ∞-stacks.