Algebraic Topology and Group Theory

We study effective methods in algebraic topology and group theory, with a view towards the computation of normal forms or bases, and the construction of small resolutions of various algebraic structures: monoids and groups (especially braid monoids and generalisations), algebras and operads, categories and higher structures, etc. The construction methods can come from combinatorial group theory (rewriting, Garside theory), combinatorial algebra (Gröbner bases, reduction operators), or homotopical and homological algebra (collapsing schemes / Morse theory, spectral sequences, Koszul duality). At the foundational level, we explore potential deep connexions between these different points of view, to unify, generalise and improve them. At the practical level, we implement libraries to compute normal forms and resolutions in various settings.

These activities are developed in immersion in the TGA (Topologie et Géométrie Algébriques) and GRG (Groupes, Représentations et Géométrie) groups of IMJ-PRG. They are part of the following funded projects :

The workshop Homology of Configuration Spaces and related topics took place in May 2023 at IMJ-PRG.

Recent related publications (2018-2023)

  • [PDF] [DOI] P. Curien, A. {DH}urić, and Y. Guiraud, “Coherent presentations of monoids with a right-noetherian Garside family,” Journal of Homotopy and Related Structures, vol. 18, pp. 115-152, 2023.
    [Bibtex] 
    @article{curien:hal-03276119,
TITLE = {{Coherent presentations of monoids with a right-noetherian Garside family}},
AUTHOR = {Curien, Pierre-Louis and {\DH}uri{\'c}, Alen and Guiraud, Yves},
URL = {https://hal.science/hal-03276119},
JOURNAL = {{Journal of Homotopy and Related Structures}},
PUBLISHER = {{Springer}},
VOLUME = {18},
PAGES = {115-152},
YEAR = {2023},
DOI = {10.1007/s40062-023-00323-4},
KEYWORDS = {Monoid ; Coherent presentation ; Higher rewriting ; Polygraph ; Artin-Tits monoid ; Garside family},
PDF = {https://hal.science/hal-03276119v4/file/coherent.pdf},
HAL_ID = {hal-03276119},
HAL_VERSION = {v4},
}

  • [PDF] Y. Guiraud, “Rewriting methods in higher algebra,” Habilitation à diriger des recherches PhD Thesis, 2019.
    [Bibtex] 
    @phdthesis{guiraud:tel-02161197,
TITLE = {{Rewriting methods in higher algebra}},
AUTHOR = {Guiraud, Yves},
URL = {https://hal.science/tel-02161197},
SCHOOL = {{Universit{\'e} Paris 7}},
YEAR = {2019},
MONTH = Jun,
KEYWORDS = {Homotopy ; Homology ; Monoid ; Artin monoid ; Associative algebra ; Higher category ; Polygraph ; Rewriting ; Squier's theorem ; Cat{\'e}gorie sup{\'e}rieure ; Polygraphe ; R{\'e}{\'e}criture ; Th{\'e}or{\`e}me de Squier ; Homotopie ; Homologie ; Coh{\'e}rence ; R{\'e}solution ; Mono{\"i}de ; Mono{\"i}de d'Artin ; Alg{\`e}bre associative},
TYPE = {Habilitation {\`a} diriger des recherches},
PDF = {https://hal.science/tel-02161197/file/hdr.pdf},
HAL_ID = {tel-02161197},
HAL_VERSION = {v1},
}

  • [PDF] [DOI] Y. Guiraud, E. Hoffbeck, and P. Malbos, “Convergent presentations and polygraphic resolutions of associative algebras,” Mathematische Zeitschrift, vol. 293, iss. 1-2, pp. 113-179, 2019.
    [Bibtex] 
    @article{guiraud:hal-01006220,
TITLE = {{Convergent presentations and polygraphic resolutions of associative algebras}},
AUTHOR = {Guiraud, Yves and Hoffbeck, Eric and Malbos, Philippe},
URL = {https://hal.science/hal-01006220},
JOURNAL = {{Mathematische Zeitschrift}},
PUBLISHER = {{Springer}},
VOLUME = {293},
NUMBER = {1-2},
PAGES = {113-179},
YEAR = {2019},
DOI = {10.1007/s00209-018-2185-z},
KEYWORDS = {Linear rewriting ; Higher-dimensional associative algebras ; Confluence and termination ; Polygraphs ; Free resolutions ; Koszulness},
PDF = {https://hal.science/hal-01006220v3/file/groebner.pdf},
HAL_ID = {hal-01006220},
HAL_VERSION = {v3},
}

  • [PDF] [DOI] Y. Guiraud and P. Malbos, “Polygraphs of finite derivation type,” Mathematical Structures in Computer Science, vol. 28, iss. 2, pp. 155-201, 2018.
    [Bibtex] 
    @article{guiraud:hal-00932845,
TITLE = {{Polygraphs of finite derivation type}},
AUTHOR = {Guiraud, Yves and Malbos, Philippe},
URL = {https://hal.science/hal-00932845},
JOURNAL = {{Mathematical Structures in Computer Science}},
PUBLISHER = {{Cambridge University Press (CUP)}},
VOLUME = {28},
NUMBER = {2},
PAGES = {155-201},
YEAR = {2018},
DOI = {10.1017/S0960129516000220},
KEYWORDS = {Low-dimensional homotopy ; Higher-dimensional categories ; Higher-dimensional rewriting ; Finite derivation type},
PDF = {https://hal.science/hal-00932845v2/file/ihdr.pdf},
HAL_ID = {hal-00932845},
HAL_VERSION = {v2},
}

Recent software

  • Garside.jl: a library for the explicit computation of a minimal resolution of Garside monoids, in Julia

Older related publications (before 2018)

  • [PDF] [DOI] P. Dehornoy and Y. Guiraud, “Quadratic normalisation in monoids,” International Journal of Algebra and Computation, vol. 26, iss. 5, pp. 935-972, 2016.
    [Bibtex] 
    @article{dehornoy:hal-01141226,
TITLE = {{Quadratic normalisation in monoids}},
AUTHOR = {Dehornoy, Patrick and Guiraud, Yves},
URL = {https://hal.science/hal-01141226},
JOURNAL = {{International Journal of Algebra and Computation}},
PUBLISHER = {{World Scientific Publishing}},
VOLUME = {26},
NUMBER = {5},
PAGES = {935-972},
YEAR = {2016},
MONTH = Aug,
DOI = {10.1142/S0218196716500399},
PDF = {https://hal.science/hal-01141226v2/file/Dip.pdf},
HAL_ID = {hal-01141226},
HAL_VERSION = {v2},
}

  • [PDF] [DOI] S. Gaussent, Y. Guiraud, and P. Malbos, “Coherent presentations of Artin monoids,” Compositio Mathematica, vol. 151, iss. 5, pp. 957-998, 2015.
    [Bibtex] 
    @article{gaussent:hal-00682233,
TITLE = {{Coherent presentations of Artin monoids}},
AUTHOR = {Gaussent, St{\'e}phane and Guiraud, Yves and Malbos, Philippe},
URL = {https://hal.science/hal-00682233},
JOURNAL = {{Compositio Mathematica}},
PUBLISHER = {{Foundation Compositio Mathematica}},
VOLUME = {151},
NUMBER = {5},
PAGES = {957-998},
YEAR = {2015},
DOI = {10.1112/S0010437X14007842},
PDF = {https://hal.science/hal-00682233v4/file/polycox.pdf},
HAL_ID = {hal-00682233},
HAL_VERSION = {v4},
}

  • [PDF] [DOI] Y. Guiraud, P. Malbos, and S. Mimram, “A Homotopical Completion Procedure with Applications to Coherence of Monoids,” in RTA – 24th International Conference on Rewriting Techniques and Applications – 2013, Eindhoven, Netherlands, 2013, pp. 223-238.
    [Bibtex] 
    @inproceedings{guiraud:hal-00818253,
TITLE = {{A Homotopical Completion Procedure with Applications to Coherence of Monoids}},
AUTHOR = {Guiraud, Yves and Malbos, Philippe and Mimram, Samuel},
URL = {https://inria.hal.science/hal-00818253},
BOOKTITLE = {{RTA - 24th International Conference on Rewriting Techniques and Applications - 2013}},
ADDRESS = {Eindhoven, Netherlands},
EDITOR = {Van Raamsdonk, Femke},
PUBLISHER = {{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik}},
SERIES = {Leibniz International Proceedings in Informatics (LIPIcs)},
VOLUME = {21},
PAGES = {223-238},
YEAR = {2013},
MONTH = Jun,
DOI = {10.4230/LIPIcs.RTA.2013.223},
KEYWORDS = {higher-dimensional rewriting ; presentation of monoid ; Knuth-Bendix completion ; Tietze transformation ; low-dimensional homotopy for monoids ; coherence},
PDF = {https://inria.hal.science/hal-00818253/file/rta13FinalLong.pdf},
HAL_ID = {hal-00818253},
HAL_VERSION = {v1},
}

  • [PDF] Y. Guiraud and P. Malbos, “Identities among relations for higher-dimensional rewriting systems,” Séminaires et congrès, vol. 26, pp. 145-161, 2013.
    [Bibtex] 
    @article{guiraud:hal-00426228,
TITLE = {{Identities among relations for higher-dimensional rewriting systems}},
AUTHOR = {Guiraud, Yves and Malbos, Philippe},
URL = {https://hal.science/hal-00426228},
JOURNAL = {{S{\'e}minaires et congr{\`e}s}},
PUBLISHER = {{Soci{\'e}t{\'e} math{\'e}matique de France}},
VOLUME = {26},
PAGES = {145-161},
YEAR = {2013},
PDF = {https://hal.science/hal-00426228v2/file/niar.pdf},
HAL_ID = {hal-00426228},
HAL_VERSION = {v2},
}

  • [PDF] [DOI] Y. Guiraud and P. Malbos, “Higher-dimensional normalisation strategies for acyclicity,” Advances in Mathematics, vol. 231, iss. 3-4, pp. 2294-2351, 2012.
    [Bibtex] 
    @article{guiraud:hal-00531242,
TITLE = {{Higher-dimensional normalisation strategies for acyclicity}},
AUTHOR = {Guiraud, Yves and Malbos, Philippe},
URL = {https://hal.science/hal-00531242},
JOURNAL = {{Advances in Mathematics}},
PUBLISHER = {{Elsevier}},
VOLUME = {231},
NUMBER = {3-4},
PAGES = {2294-2351},
YEAR = {2012},
DOI = {10.1016/j.aim.2012.05.010},
KEYWORDS = {rewriting ; polygraphic resolution ; homology of small categories ; identities among relations},
PDF = {https://hal.science/hal-00531242v3/file/ktheory.pdf},
HAL_ID = {hal-00531242},
HAL_VERSION = {v3},
}

  • [PDF] [DOI] Y. Guiraud and P. Malbos, “Coherence in monoidal track categories,” Mathematical Structures in Computer Science, vol. 22, iss. 6, pp. 931-969, 2012.
    [Bibtex] 
    @article{guiraud:hal-00470795,
TITLE = {{Coherence in monoidal track categories}},
AUTHOR = {Guiraud, Yves and Malbos, Philippe},
URL = {https://hal.science/hal-00470795},
JOURNAL = {{Mathematical Structures in Computer Science}},
PUBLISHER = {{Cambridge University Press (CUP)}},
VOLUME = {22},
NUMBER = {6},
PAGES = {931-969},
YEAR = {2012},
DOI = {10.1017/S096012951100065X},
KEYWORDS = {Coherence ; Monoidal category ; Higher-dimensional category ; Rewriting ; Polygraph},
PDF = {https://hal.science/hal-00470795v2/file/cohe.pdf},
HAL_ID = {hal-00470795},
HAL_VERSION = {v2},
}

  • [PDF] [DOI] Y. Guiraud and P. Malbos, “Higher-dimensional categories with finite derivation type,” Theory and Applications of Categories, vol. 22, iss. 18, pp. 420-478, 2009.
    [Bibtex] 
    @article{guiraud:hal-00326974,
TITLE = {{Higher-dimensional categories with finite derivation type}},
AUTHOR = {Guiraud, Yves and Malbos, Philippe},
URL = {https://hal.science/hal-00326974},
JOURNAL = {{Theory and Applications of Categories}},
PUBLISHER = {{Mount Allison University}},
VOLUME = {22},
NUMBER = {18},
PAGES = {420-478},
YEAR = {2009},
DOI = {10.48550/arXiv.0810.1442},
KEYWORDS = {$n$-category ; rewriting ; polygraph ; finite derivation type ; low-dimensional topology},
PDF = {https://hal.science/hal-00326974v2/file/ntdf.pdf},
HAL_ID = {hal-00326974},
HAL_VERSION = {v2},
}

  • [PDF] [DOI] G. Bonfante and Y. Guiraud, “Polygraphic programs and polynomial-time functions,” Logical Methods in Computer Science, vol. 5, iss. 2:14, pp. 1-37, 2009.
    [Bibtex] 
    @article{bonfante:inria-00122932,
TITLE = {{Polygraphic programs and polynomial-time functions}},
AUTHOR = {Bonfante, Guillaume and Guiraud, Yves},
URL = {https://inria.hal.science/inria-00122932},
JOURNAL = {{Logical Methods in Computer Science}},
PUBLISHER = {{Logical Methods in Computer Science Association}},
VOLUME = {5},
NUMBER = {2:14},
PAGES = {1-37},
YEAR = {2009},
DOI = {10.2168/LMCS-5(2:14)2009},
KEYWORDS = {Polygraph ; Polygraphic program ; Polygraphic interpretation ; Computability ; Complexity ; Polynomial time},
PDF = {https://inria.hal.science/inria-00122932v3/file/polypoly.pdf},
HAL_ID = {inria-00122932},
HAL_VERSION = {v3},
}

  • [PDF] [DOI] G. Bonfante and Y. Guiraud, “Intensional properties of polygraphs,” in 4th International Workshop on Computing with Terms and Graphs – TERMGRAPH 2007, Braga, Portugal, 2007.
    [Bibtex] 
    @inproceedings{bonfante:inria-00129391,
TITLE = {{Intensional properties of polygraphs}},
AUTHOR = {Bonfante, Guillaume and Guiraud, Yves},
URL = {https://inria.hal.science/inria-00129391},
BOOKTITLE = {{4th International Workshop on Computing with Terms and Graphs - TERMGRAPH 2007}},
ADDRESS = {Braga, Portugal},
SERIES = {Electronic Notes in Theoretical Computer Science},
VOLUME = {203(1):65-77},
YEAR = {2007},
MONTH = Mar,
DOI = {10.1016/j.entcs.2008.03.034},
KEYWORDS = {Polygraph ; Polygraphic program ; Polygraphic interpretation ; Computability ; Complexity ; Polynomial time},
PDF = {https://inria.hal.science/inria-00129391v3/file/termgraph.pdf},
HAL_ID = {inria-00129391},
HAL_VERSION = {v3},
}

  • [PDF] [DOI] Y. Guiraud, “The three dimensions of proofs,” Annals of Pure and Applied Logic, vol. 141(1-2), pp. 266-295, 2006.
    [Bibtex] 
    @article{guiraud:hal-00092212,
TITLE = {{The three dimensions of proofs}},
AUTHOR = {Guiraud, Yves},
URL = {https://hal.science/hal-00092212},
JOURNAL = {{Annals of Pure and Applied Logic}},
PUBLISHER = {{Elsevier Masson}},
VOLUME = {141(1-2)},
PAGES = {266-295},
YEAR = {2006},
DOI = {10.1016/j.apal.2005.12.012},
PDF = {https://hal.science/hal-00092212/file/cos.pdf},
HAL_ID = {hal-00092212},
HAL_VERSION = {v1},
}

  • [PDF] [DOI] Y. Guiraud, “Two polygraphic presentations of Petri nets,” Theoretical Computer Science, vol. 360(1-3), pp. 124-146, 2006.
    [Bibtex] 
    @article{guiraud:hal-00092209,
TITLE = {{Two polygraphic presentations of Petri nets}},
AUTHOR = {Guiraud, Yves},
URL = {https://hal.science/hal-00092209},
JOURNAL = {{Theoretical Computer Science}},
PUBLISHER = {{Elsevier}},
VOLUME = {360(1-3)},
PAGES = {124-146},
YEAR = {2006},
DOI = {10.1016/j.tcs.2006.02.015},
KEYWORDS = {Polygraph ; n-Category ; Petri net},
PDF = {https://hal.science/hal-00092209/file/rdp.pdf},
HAL_ID = {hal-00092209},
HAL_VERSION = {v1},
}

  • [PDF] [DOI] Y. Guiraud, “Termination orders for 3-dimensional rewriting,” Journal of Pure and Applied Algebra, vol. 207(2), pp. 341-371, 2006.
    [Bibtex] 
    @article{guiraud:hal-00092204,
TITLE = {{Termination orders for 3-dimensional rewriting}},
AUTHOR = {Guiraud, Yves},
URL = {https://hal.science/hal-00092204},
JOURNAL = {{Journal of Pure and Applied Algebra}},
PUBLISHER = {{Elsevier}},
VOLUME = {207(2)},
PAGES = {341-371},
YEAR = {2006},
DOI = {10.1016/j.jpaa.2005.10.011},
KEYWORDS = {Polygraph ; n-Category ; Rewriting ; Term rewriting ; Termination ; Confluence ; Termination order},
PDF = {https://hal.science/hal-00092204/file/erm.pdf},
HAL_ID = {hal-00092204},
HAL_VERSION = {v1},
}

  • [PDF] [DOI] Y. Guiraud, “Termination orders for 3-polygraphs,” Comptes rendus de l’Académie des sciences. Série I, Mathématique, vol. 342(4), pp. 219-222, 2006.
    [Bibtex] 
    @article{guiraud:hal-00092196,
TITLE = {{Termination orders for 3-polygraphs}},
AUTHOR = {Guiraud, Yves},
URL = {https://hal.science/hal-00092196},
JOURNAL = {{Comptes rendus de l'Acad{\'e}mie des sciences. S{\'e}rie I, Math{\'e}matique}},
PUBLISHER = {{Elsevier}},
VOLUME = {342(4)},
PAGES = {219-222},
YEAR = {2006},
DOI = {10.1016/j.crma.2005.12.019},
KEYWORDS = {Polygraph ; n-Category ; Rewriting ; Termination ; Termination order},
PDF = {https://hal.science/hal-00092196/file/cras1.pdf},
HAL_ID = {hal-00092196},
HAL_VERSION = {v1},
}

Older software

  • Catex: a Latex tool for string diagrams, in OCaml
  • Cox: functions to compute coherent presentations of Artin monoids, in Python
  • dCat: a prototype for automatic computation of complexity bounds of polygraphic programs, in OCaml
  • Rewr: a prototype for higher-dimensional rewriting, in OCaml

 

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