Mathematical Background

My mathematical culture has been built through a long process of detailed reconstruction and personal appropriation of mathematical theories.

Before specializing in higher category theory, I spent many years studying a broad range of mathematical subjects in great depth. My approach has always been to return to definitions, constructions, proofs, and logical dependencies, and to understand each theory as explicitly as possible. I have often rewritten entire developments in my own notes, sometimes over hundreds of pages, in order to make every step transparent.

A substantial part of this work concerned analysis and differential geometry. I studied differential calculus in detail, including the mean value theorem and its refinements, differential equations, parameter-dependent integrals, and the foundations of integration theory. I also devoted considerable time to measure theory and Lebesgue integration, comparing several sources and reconstructing the subject in a form that made its internal architecture fully explicit.

Differential geometry was another major area of study. I worked carefully through the various constructions of tangent spaces, differential forms, integration on manifolds, and Stokes' theorem. More generally, I spent several years studying smooth manifolds and fibre bundles, with particular attention to the way local differential data are assembled into global geometric structures.

This interest in local-to-global constructions naturally led me to sheaf theory. I studied in detail the construction of the sheaf associated with a presheaf and the general mechanisms of gluing algebraic and geometric data. This later extended to stacks, whose conceptual structure I first encountered through categorical and topos-theoretic literature.

Another important part of my mathematical formation was homological and algebraic topology. I worked through singular cohomology, its basic constructions, and the standard cohomological operations, including cup and cap products. These subjects played an important role in the development of my intuition concerning higher-dimensional algebraic structures.

In parallel, I undertook a systematic study of category theory. At one stage I set myself the personal challenge of understanding, in detail, most of the principal constructions and theorems of Mac Lane's Categories for the Working Mathematician. This included adjunctions, monads, Eilenberg-Moore and Kleisli categories, Kan extensions, ends and coends, cofinality, and related universal constructions.

I also studied Beck's monadicity theorem in detail, as well as localization of categories, calculus of fractions, accessible categories, locally presentable categories, and various forms of factorization systems.

Transfinite constructions have occupied an important place in this work. I studied the small object argument with particular care, both in its classical form and in more algebraic refinements. I also examined transfinite induction and transfinite recursion from foundational principles, and later their role in categorical and homotopical constructions.

My interest in foundations led me to logic and set theory. Starting from the axioms of Zermelo-Fraenkel set theory, I studied well-orderings, ordinal numbers, transfinite recursion and induction, ordinal arithmetic, Hartogs' construction, cardinal numbers, and cardinal arithmetic. The purpose was not simply to use these results, but to understand how the basic architecture of set theory is actually built.

Over time, these different lines of study converged with my work in abstract homotopy theory. I studied model categories, localization, factorization systems, the small object argument, bar constructions, and categorical methods underlying homotopical algebra.

For around ten years, I also taught mathematics at the level of the French classes préparatoires, in particular mathematics corresponding roughly to advanced undergraduate level. This teaching experience reinforced my habit of returning to elementary and intermediate results with the same degree of care that one brings to more advanced mathematics.

What connects all these subjects is not primarily their diversity, but a particular way of working. I tend to approach a theory by dismantling it into its basic components: definitions, lemmas, constructions, hidden assumptions, and logical transitions. I then rebuild it step by step until its internal structure becomes completely visible.

This slow and detailed process of appropriation has been one of the main sources of my mathematical intuition. It also strongly influences my current research in higher category theory, where questions of coherence, recursive structure, universal constructions, and higher-dimensional algebra require precisely this kind of attention to architecture.

The texts collected in this section are therefore not intended as research publications in the usual sense. They are records of mathematical understanding: detailed reconstructions, personal expositions, proofs, calculations, and conceptual reorganizations produced over many years of study.