Master 2 Advanced course on Categorical and operadic Rewriting, Spring 2020

Course

The class takes place on Wednesdays.

  • 29/01/2020- 10am-noon: Séminaire 2
  • 05/02/2020- 8:30am-10:30am: Séminaire 1
  • 12/02/2020- 10am-noon: Séminaire 2
  • 19/02/2020- 9am-11am: 125
  • 26/02/2020- 9am-11am: 125
  • 11/03/2020- 10am-noon: 125

    • Course

    The first part of the course is given by Philippe Malbos inside the program Representation Theory, from Schubert Calculus to Rewriting
    Lecture notes of the second part of the course on Operadic rewriting (in french):Version of March 7.
    Tentative Program
  • Lecture 1: Introduction to operads (chap 1), Shuffles and examples of Shuffle Trees (chap 2)

    • Motivation, Definitions of S-modules and symmetric operads (using bilinear operadic compositions). Notation of composition with trees. Examples of endomorphism operads, operads coming from associative algebras. Def of morphisms of operads and algebras over operads. Examples of operads Com and Ass. Comments on alternative definitions: monoidal and monadic definitions, colored operads. Preliminary on shuffles and first example of shuffle trees (corresponding to a shuffle composition).


  • Lecture 2: Shuffle Trees, Free Shuffle Operads, operadic ideals (chap 2).

    • Definitions of planar trees and shuffle trees. Definition of shuffle subtrees and quotient shuffle trees. Definition of the free shuffle operad functor and shuffle monomials. Monad structure on the free shuffle operad functor via substitution of trees. Examples of substitution of a subtree in a quotient tree. Examples of binary shuffle compositions. Definition of shuffle operadic ideals.

  • Lecture 3: Monomial orders. Divisibility in operads. Operadic rewriting and normal forms (Chap 3)

    • Forgetful functor from symmetric operads to shuffle operads. Examples of presentations: shuffle version of the Three Graces: Com,Lie,Ass. Chapter 3: Definitions of operadic monomial orders. Paths-Permutation data invariant of a monomial. Example of degPathPerm order on free shuffle operads associated to a monoidal monomial order. Divisibility and divisor in free shuffle operad. Definition of reduction and corresponding abstract ARS on the reduced free shuffle operad. Preparation of the generating relations by linear normalization.


  • Lecture 4: Grobner Basis in terms of operadic rewriting. (chap 3)

    • Termination of ARS(S,<). Operadic Gröbner basis. Definition. Equivalence to having operadically reduced monomials as basis of the quotient shuffle operad. Equivalence to the convergence of ARS(S,<).


  • Lecture 5: Critical branching lemma. Examples of operadic Grobner bases (chap 4).

    • S-polynomials. Peiffer and critical branchings. Buchberger algorithm. Examples: Com, Lie (Melançon-Reutenauer basis, Dynkin basis), Ass, NsOp (coloured operad of non-symetric operads and brief description of the relation of normal forms to planar trees)


  • Lecture 6: Koszul duals, Koszul Shuffle operads (Chap 5)

    • Differential graded operads. Quadratic presentation and Koszul duals operad. Examples of Koszul duals (same as for Grobner bases). Cobar complex of an operad (and its weight grading). Koszul operads: Hoffbeck-Dotsenko-Khoroshkin criteria by quadratic Grobner bases (proof of monomial case, and reduction to monomial case by a spectral sequence).


    List of articles for evaluation



  • Vladimir Dotsenko and Pedro Tamaroff, Endofunctors and Poincaré-Birkhoff-Witt theorems, arXiv:1804.06485, Int. Math. Res. Notices, Vol. 2020: article ID rnz369, 21 pages.

    • A general categorical result enabling to describe the freest P-algebra of a Q-algebra for two related operads P-Q by a functorial isomorphism. The general categorical result requires a freeness assumption proven with techniques of the class in section 4. (the paper is a little far from the class, but good if you like categories. It may use in examples the last paper of the list.).


  • Vladimir Dotsenko, A Quillen adjunction between algebras and operads, Koszul duality, and the Lagrange inversion formula, arXiv:1606.08222, Int. Math. Res. Notices, Vol. 2017: 21 pages.

    • Description of a link between algebraic and operadic contexts enabling to import counterexamples for algebras to operads.


  • Vladimir Dotsenko and James Griffin, Cacti and filtered distributive laws, arXiv:1109.5345, Algebraic and Geometric Topology, 14 (2014), issue 6, 3185-3225.

    • Section 5 is an interesting application of ideas of the class (for a specific form of ordering which appear often in applications) Many examples are treated, including a more sophisticated one described in the first 4 sections (From the viewpoint of the class, one should focus first on the 2 last somehow sketchy sections).


  • Vladimir Dotsenko and Anton Khoroshkin, Quillen homology for operads via Gröbner bases, arXiv:1203.5053, Documenta Mathematica, 18 (2013), 707-747.

    • Long and advanced, a continuation of the last lecture of the class (beyond quadratic Gröbner bases): Operadic version of Anick's resolution.


  • Vladimir Dotsenko, Freeness theorems for operads via Gröbner bases, arXiv:0907.4958, Séminaires et Congrès, Volume 26 (2011), 61-76

    • A short and direct application of the class (focusing on computation of bases instead of homological applications). Short proofs requiring to fill in details, especially in examples.


    References

    Books


  • M. Bremner et V. Dotsenko, Algebraic Operads : An Algorithmic Companion Chapman & Hall / CRC Press, 2016, 365 pages Version from the first author's website
    • This book is very close to this class, it introduces Grobner basis in details, but it does not use rewriting explicitly (of course the part on Grobner basis can be formulated in terms of rewriting as we will do), it contains many supplementary exercises and examples.
  • J- L Loday et B. Vallette, Algebraic Operads, Springer-Verlag Berlin 2012 Version from the second author's website
    • This book is an encyclopedic treatment of results about Koszul duality for algebras and operads. The operadic part is hard to read separately from the advised reading order. Operadic rewriting is treated in Chapter 8.

    Papers

    Among the papers quoted in the lecture notes bibliography, some are more extensively used in this class.
  • V. Dotsenko et A. Khoroshkin, Gröbner basis for Operads Duke Math. J. 153, 2 (2010) pp 363–396.Official link
  • V. Ginzburg and M. Kapranov , Koszul duality for operads, Duke Math. J. 76 (1994), 203 – 272Official link

    • Contact

    Je vous invite à consulter régulièrement cette page, mise à jour régulièrement, pour en savoir plus.

    N'hésitez pas à me contacter par e-mail.

    mèl : dabrowski at math . univ-lyon1 . fr