ANR CATHRE Categories, Homotopy and Rewriting

ANR-13-BS02-0005-02

Presentation

The CATHRE project is rooted in the works of Squier, Anick and others on homological properties of presentations of monoids and algebras. Squier showed in particular that if a monoid $$\mathscr{M}$$ can be presented by a finite, confluent and terminating rewriting system, then its third integral homology group $$\mathrm{H}_3(\mathscr{M},\mathbb{Z})$$ is finitely generated as an abelian group. He later refined this result by showing that, under the same hypotheses, $$\mathscr{M}$$ has finite derivation type, a property of homotopical nature. We believe that these are not isolated results, but rather ought to be part of a wide homotopical theory of rewriting, encompassing words, terms, linear combinations of terms, as well as diagrams of various shapes. The project aims at developing such a theory, together with computational tools based on it, starting from the following two observations:

• all objects of study, such as words, terms or rewriting rules can be expressed very naturally in the language of strict, globular, higher-dimensional categories;
• the category of these higher categories has a non-trivial homotopy structure.
Even in the simple case of monoids, arbitrary high dimensions appear in the process of resolution: the computation of homological or homotopical invariants involves not only 1-dimensional words and 2-dimensional rewriting paths, but also 3-dimensional deformations between paths and so on. Our approach is supported by strong results already obtained in this direction, namely
• the analysis of numerous examples of higher-dimensional rewriting
• the discovery of a natural model structure on strict higher categories.
The CATHRE project intends to develop these two lines of research with the aim of implementing a computational system which is able to handle various kinds of algebraic structures and, moreover, to explore three main fields of application
• in symbolic algebra, for computation in algebras and operads,
• in combinatorial group theory, for computation in groups involved in algebra and geometry,
• in term rewriting, for functional programming.

Meetings

• Next meeting: Dec. 2015 or Jan. 2016 at IMT, Toulouse
• 12 June 2015: project meeting at LIX, Palaiseau
• 2-3 Feb. 2015: project meeting at ICJ, Lyon
• 29 Sept. – 3 Oct. 2014: workshop Algebras, Operads and Rewriting at ICJ, Saint-Étienne
• 22-23 May 2014: project meeting at IHP, Paris
• 22 Jan. 2014: kick-off meeting at ICJ, Lyon

Members

Paris partner

Permanent researchers:

PhD students:
• Cyrille Chenavier
• Maxime Lucas

Lyon partner

Permanent researchers:

PhD students:

• Matteo Acclavio
• Clément Alleaume
• Norah Hage

Publications and preprints

List of publications and preprints of the CATHRE project on Hal
1. Clément Alleaume, Rewriting in the category of Bott-Samelson bimodules, Higher-Dimensional Rewriting and Applications 2015
2. Dimitri Ara et Georges Maltsiniotis, Le type d'homotopie de la $$\infty$$-catégorie associée à un complexe simplicial, preprint 2015
3. Emily Burgunder, Pierre-Louis Curien and Maria Ronco, Free algebraic structures on the permutohedra, preprint 2015
4. Cyrille Chenavier, Confluence algebras and acyclicity of the Koszul complex, preprint 2015
5. Florence Clerc and Samuel Mimram, Presenting a category modulo a rewriting system, Rewriting Techniques and Applications 2015
6. Pierre-Louis Curien and Jovana Obradović, On the various definitions of cyclic operads, Category Theory 2015
7. Patrick Dehornoy and Yves Guiraud, Quadratic normalisation in monoids, preprint 2015
8. Stéphane Gaussent, Yves Guiraud and Philippe Malbos, Coherent presentations of Artin monoids, Compos. Math. 2015
9. Yves Guiraud, Eric Hoffbeck and Philippe Malbos, Confluence of linear rewriting and homology of algebras, International Workshop on Confluence 2014
10. Yves Guiraud, Eric Hoffbeck and Philippe Malbos, Linear polygraphs and Koszulity of algebras, preprint 2014
11. Yves Guiraud and Philippe Malbos, Polygraphs of finite derivation type, Math. Structures Comput. Sci. (to appear)
12. Norah Hage, Finite convergent presentation of plactic monoid for type C, preprint 2014
13. Eric Hoffbeck and Christine Vespa, Leibniz homology of Lie algebras as functor homology, J. Pure Appl. Algebra 2015
14. Maxime Lucas, A coherence theorem for pseudonatural transformation, preprint 2015
15. Joan Millès, Complex manifolds as families of homotopy algebras, preprint 2014
16. Samuel Mimram, Presenting finite posets, Computing with Terms and Graphs 2014
17. Samuel Mimram, Towards 3-dimensional rewriting theory, Log. Methods Comput. Sci. 2014
18. Jovana Obradović, The Bénabou-Roubaud monadic descent theorem via string diagrams, Higher-Dimensional Rewriting and Applications 2015