Section: Research Program
Effective higher-dimensional algebra
Higher-dimensional algebra
Like ordinary categories, higher-dimensional categorical structures originate in algebraic topology. Indeed,
In the last decades, the importance of higher-dimensional categories has grown fast, mainly with the new trend of categorification that currently touches algebra and the surrounding fields of mathematics. Categorification is an informal process that consists in the study of higher-dimensional versions of known algebraic objects (such as higher Lie algebras in mathematical physics [65]) and/or of “weakened” versions of those objects, where equations hold only up to suitable equivalences (such as weak actions of monoids and groups in representation theory [80]).
Since a few years, the categorification process has reached logic, with the introduction of homotopy type theory. After a preliminary result that had identified categorical structures in type theory [94], it has been observed recently that the so-called “identity types” are naturally equiped with a structure of
Higher-dimensional rewriting
Higher-dimensional categories are algebraic structures that contain, in essence, computational aspects. This has been recognised by Street [112], and independently by Burroni [71], when they have introduced the concept of computad or polygraph as combinatorial descriptions of higher categories. Those are directed presentations of higher-dimensional categories, generalising word and term rewriting systems.
In the recent years, the algebraic structure of polygraph has led to a new theory of rewriting, called higher-dimensional rewriting, as a unifying point of view for usual rewriting paradigms, namely abstract, word and term rewriting [99], [104], [86], [87], and beyond: Petri nets [89] and formal proofs of classical and linear logic have been expressed in this framework [88]. Higher-dimensional rewriting has developed its own methods to analyse computational properties of polygraphs, using in particular algebraic tools such as derivations to prove termination, which in turn led to new tools for complexity analysis [68].
Squier theory
The homotopical properties of higher categories, as studied in mathematics, are in fact deeply related to the computational properties of their polygraphic presentations. This connection has its roots in a tradition of using rewriting-like methods in algebra, and more specifically in the work of Anick [63] and Squier [111], [110] in the 1980s: Squier has proved that, if a monoid
In particular, the computational content of Squier's proof has led to a constructive methodology to produce, from a convergent presentation, coherent presentations and polygraphic resolutions of algebraic structures, such as monoids [14] and algebras [31]. A coherent presentation of a monoid