Koszul Duality
V5D1 - Advanced Topics in Topology. Course page on BASIS
Time and Location
Wednesday, 12:15 - 14:00 in Hörsaal XV
Thursday 12:15 - 14:00 in Hörsaal XVI
Institute of Geodesy and Geoinformation
Nußallee 17, 53115 Bonn. See here for a map and for descriptions of the lecture halls.
Oral exam
1. Exam periode: Monday 17.02 - Wednesday 19.02
2. Exam periode: Monday 24.02 - Wednesday 26.02
Course description
Koszul duality is a concept from homological algebra that establishes a relationship between augmented associative algebras and augmented co-associative coalgebras, providing a comparison functor between the derived category of an algebra and the derived category of its Koszul dual algebra. Operadic Koszul duality, introduced by Ginzburg and Kapranov, extends this notion to the realm of the theory of operads, which has found many applications in deformation theory, algebraic topology and category theory.
In the course we will begin with the classical algebraic Koszul duality in differential graded setting. Then we will introduce operadic Koszul duality in ∞-categorical languages, beginning with some prerequisites on the theory of ∞-category and higher algebras. We will complement the abstract theories with many applications, such as rational homotopy theory and deformation theory in characteristic 0. If time permits, we will discuss deformation theory in positive characteristic towards the end of the course.
Schedule
Algebraic Koszul duality
- 09.10 - Introduction and some prerequisites on homological algebras.
- 10.10 - Basic notions of Koszul algebras.
- 16.10 - Lecture by Pete Teichner: Q&A Session.
- 17.10 - Guest lecture by Connor Malin.
- 23.10 - Examples of Koszul algebras and labelled basis.
- 24.10 - Induced generators and relations of the cohomology of a quadratic Koszul algebra: State the theorem.
- 30.10 - Induced generators and relations of the cohomology of a quadratic Koszul algebra: Proof of the theorem.
- 31.10 - Introduction to Steenrod square.
- 06.11 - The Steenrod algebra is Koszul.
- 07.11 - Koszul resolution and Koszul duality.
13.11 - 14.11 - No lectures because of the Workshop on Unstable Homotopy Theory
The theory of operads
- 20.11 - Definition of operads, algebras over an operads and examples.
- 21.11 - Operdas, symmetric seqneces and the category of operators.
The language of infinity categories
- 27.11 - Motivations and definition of ∞-categories.
- 28.11 - The language of ∞-categories.
- 04.12 - No lecture due to Dies Academicus.
- 05.12 - Lecture by Thomas Blom on Straightening and Unstraightening.
The theory of ∞-operads
Infinity categorical operadic Koszul duality
Applications
Recourses
I have consulted the following lecture courses, in order to design my course on Koszul duality.
- D. Lukas B. Brantner. Topics in Koszul duality. 2019. Course website.
- T. Dyckerhoff. Introduction to higher category theory. 2018, 2020.
- O. Gwilliam and C. Scheimbauer. Derived deformation theory and Koszul duality. 2016. Course website.
- K. Wickelgren. A user's guide to infinity categories. 2023 Course website.
- BIREP Summer School on Koszul Duality. A user's guide to infinity categories. 2015. Lecture notes.
Reference
On algebraic Koszul duality
- J.C. Moore. Differential homological algebra. Actes Congr. internat. Math, 1, 335-339 (1971).
- S. B. Priddy. Koszul resolutions and the Steenrod algebra. Bull. Am. Math. Soc. 76 (1969), pp. 834--839.
- S. B. Priddy. Koszul resolutions. Trans. Am. Math. Soc. 152 (1970), pp. 39--60.
- A. Polishchuk and L. Positselsk., Quadratic Algebras. American Mathematical Society (AMS) (2005).
On Koszul duality of module categories
- A. Beilinson et al.. Koszul duality patterns in representation theory. J. Am. Math. Soc. 9, No. 2, pp. 473--527 (1996).
- I. N. Bernstein et al. Algebraic bundles over ℙn and problems of linear algebra. Funct. Anal. Appl. 12, pp. 212--214 (1979).
- G. Williamson. Koszul duality and applications in representation theory. Available online.
On the theory of operads
- B. Fresse. Homotopy of operads and Grothendieck-Teichmüller groups. Part 1: The algebraic theory and its topological background. American Mathematical Society (AMS), Providence, RI (2017).
- G. Heuts and I. Moerdijk. Simplicial and dendroidal homotopy theory. Springer, Cham (2022).
- J. Lurie. Higher algebras. Available online.
- Y. Harpaz. Little cube algebras and factorization homology. Lecture notes.
On the theory of operadic Koszul duality
- V. Ginzburg and M. Kapranov. Koszul duality for operads. Duke Math. J. 76.1 (1994), pp. 203–-272.
- J. Lurie. Higher algebras. Available online.
On self Koszul duality of the En-operads
- E. Getzler and J. D. S. Jones. Operads, homotopy algebra and iterated integrals for double loop spaces. Preprint. Available at arXiv (1994).
- B. Fresse. Koszul duality of operads and homology of partition posets. Contemp. Math. 346, 115--215 (2004).
- M. Ching and P. Salvatore. Koszul duality for topological En-operads. Proc. Lond. Math. Soc. (3) 125, No. 1, 1--60 (2022).
On higher categories and higher algebras
- T. Dyckerhoff. Introduction to higher category theory. Lecture notes. Available online.
- M. Land, Introduction to infinity-categories. Birkhäuser, Cham. (2021).
- J. Lurie. Higher algebras. Available online.
- J. Lurie. Higher topos theory. Princeton University Press, Princeton, NJ (2009).
On homological algebras
- H. Cartan and S. Eilenberg. Homological Algebra. Paperback ed. Princeton University Press (1999).
- C. A. Weibel. An Introduction to Homological Algebra. Cambridge University Press (1994).
On the Bar construction
- J. F. Adams. On the cobar construction Proc. Natl. Acad. Sci. USA 42, pp. 409--412 (1956).
- N. Y. J. H. Gedara. On Koszul duality between polynomial and exterior algebras. PhD Thesis. Available online.
- M. Rivera. Adams' cobar construction revisited. Doc. Math. 27, pp. 1213--1223 (2022).
- R. Zhang. Two sided bar construction. Available online.
Prerequisites
The course is aimed at Master students with a focus on algebraic topology and algebra. Prerequisites include some basic knowledge of homological algebras and the content of the bachelor courses Topology I and Topology II in Bonn.