Section: Research Program

Effective higher-dimensional algebra

Higher-dimensional algebra

Like ordinary categories, higher-dimensional categorical structures originate in algebraic topology. Indeed, $\infty$-groupoids have been initially considered as a unified point of view for all the information contained in the homotopy groups of a topological space $X$: the fundamental $\infty$-groupoid $\Pi \left(X\right)$ of $X$ contains the elements of $X$ as 0-dimensional cells, continuous paths in $X$ as 1-cells, homotopies between continuous paths as 2-cells, and so on. This point of view translates a topological problem (to determine if two given spaces $X$ and $Y$ are homotopically equivalent) into an algebraic problem (to determine if the fundamental groupoids $\Pi \left(X\right)$ and $\Pi \left(Y\right)$ are equivalent).

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 [43]) 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 [54]).

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 [69], it has been observed recently that the so-called “identity types” are naturally equiped with a structure of $\infty$-groupoid: the 1-cells are the proofs of equality, the 2-cells are the proofs of equality between proofs of equality, and so on. The striking ressemblance with the fundamental $\infty$-groupoid of a topological space led to the conjecture that homotopy type theory could serve as a replacement of set theory as a foundational language for different fields of mathematics, and homotopical algebra in particular.

Higher-dimensional rewriting

Higher-dimensional categories are algebraic structures that contain, in essence, computational aspects. This has been recognised by Street [86], and independently by Burroni [47], 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 [73], [79], [60], [61], and beyond: Petri nets [63] and formal proofs of classical and linear logic have been expressed in this framework [62]. 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 [46].

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 [41] and Squier [85], [84] in the 1980s: Squier has proved that, if a monoid $M$ can be presented by a finite, terminating and confluent rewriting system, then its third integral homology group $H_3(M,\mathbb{Z})$ is finitely generated and the monoid $M$ has finite derivation type (a property of homotopical nature). This allowed him to conclude that finite convergent rewriting systems were not a universal solution to decide the word problem of finitely generated monoids. Since then, Yves Guiraud and Philippe Malbos have shown that this connection was part of a deeper unified theory when formulated in the higher-dimensional setting [9], [10], [66], [67], [68].

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 [9] and algebras [65]. A coherent presentation of a monoid $M$ is a 3-dimensional combinatorial object that contains not only a presentation of $M$ (generators and relations), but also higher-dimensional cells, each of which corresponding to two fundamentally different proofs of the same equality: this is, in essence, the same as the proofs of equality of proofs of equality in homotopy type theory. When this process of “unfolding” proofs of equalities is pursued in every dimension, one gets a polygraphic resolution of the starting monoid $M$. This object has the following desirable qualities: it is free and homotopically equivalent to $M$ (in the canonical model structure of higher categories [74], [42]). A polygraphic resolution of an algebraic object $X$ is a faithful formalisation of $X$ on which one can perform computations, such as homotopical or homological invariants of $X$. In particular, this has led to new algorithms and proofs in representation theory [7], and in homological algebra [64], [65].