Philippe Malbos

My research studies algebraic rewriting from a homotopical perspective. Algebraic rewriting provides a relational model of computation in which computation is expressed through local transformation rules acting on syntactic, algebraic, diagrammatic, operadic, or categorical expressions.

More specifically, I work in the framework of higher-dimensional rewriting, where rewriting systems are represented by multidimensional polygraphs that capture both the syntax of terms and the homotopical structure of rewriting.

In ongoing joint work with Georg Struth, we study higher-dimensional rewriting systems from the perspective of algebraic formalization. In this direction, together with Eric Goubault and Cameron Calk, we have developed constructive methods for proving coherence in algebraic structures and identified algebraic structures underlying the formalization of rewriting systems in proof assistants. With Tanguy Massacrier, we have proposed a cubical formulation of higher-dimensional rewriting, and a single-set axiomatisation of cubical ω-categories.

I also study higher-dimensional algebraic rewriting as a natural framework for linear rewriting in algebras and operads, as well as for diagrammatic rewriting in linear monoidal categories.

Previously, in joint work with Yves Guiraud, we developed a homotopical approach to higher-dimensional rewriting based on normalization strategies for polygraphs, allowing acyclicity proofs for higher categories by rewriting. In low dimensions, we applied this approach to study the coherence of Artin monoids, in collaboration with Stéphane Gaussent.

In collaboration with Dimitri Ara, Albert Burroni, Yves Guiraud, François Métayer, and Samuel Mimram, we published the monograph Polygraphs: from rewriting to higher categories in the London Mathematical Society Lecture Note Series. The book is also available on arXiv.