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. Room 4.009, Mathematik Zentrum (Office of Dr. Jack Davies)
2. Exam periode: Monday 24.03 - Wednesday 26.03
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.
Handout and exercises
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 (The content of this lecture is not relevant for the exam).
- 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
- 11.12 - Basic notion of Lurie's theory of ∞-operads: Definitions, examples, monoid objects and algbera object over an ∞-operads.
- 12.12 - Associative ∞-operads, left-module ∞-operads, monad and monadic adjunctions, presentable ∞-categories.
- 18.12 - Adjoint functor theorems, symmetric sequences, Day convolution, ∞-operads with values in a presentable symmetric monoidal ∞-categories.
- 19.12 - Constructing composition product monoidal structure on symmetric sequences, the theory of ∞-operads with values in a presentable symmetric monoidal ∞-categories (continued).
08.01 - 09.01 - No lectures due to the flu
Infinity categorical operadic Koszul duality
- 15.01 - Coalgebras over an ∞-cooperad, comonad and comonadic adjunctions, Bar-cobar adjunction between ∞-operads and ∞-cooperads, the indecomposable functor from algebras over an ∞-operads to conilpotent diveded power coalgebras over the Koszul dual ∞-cooperad.
- 16.01 - The Bar construction on an augmented associative algebra object, proof the statement about Koszul duality on the level of algebras.
- 22.01 - Twisted arrow category, statement of Bar--coBar Koszul duality using twisted arrow category.
- 23.01 - Koszul duality between algebras over an ∞-operad and the coalgebras over the Koszul dual ∞-cooperad. Stable ∞-category. Smash product on the ∞-category of spectra. Example of ∞-operads. Commutative ∞-operads. Linear dual of an ∞-cooperad is an ∞-operad. Spectral Lie ∞-operads.
Applications
- 29.01 - Spectral Lie algebras in spectra and chain complexes, relation to differential graded Lie algebras, basic notion of rational homotopy theory, Whitehead products, algebraic model for rational homotopy theory, proof of cocommutative coalgebra model for rational homotopy types.
- 30.01 - Sketch of the proof of spectral Lie algebra model for rational homotopy types, Lie algebra model of rational spheres, computation of rational homotopy groups of spheres.
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. 2015. Lecture notes.
Reference
On algebraic Koszul duality
- J.C. Moore. Differential homological algebra. Actes Congr. internat. Math, 1, 335-339 (1971).
- A. Polishchuk and L. Positselsk., Quadratic Algebras. American Mathematical Society (AMS) (2005).
- 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.
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).
- Y. Harpaz. Little cube algebras and factorization homology. Lecture notes.
- G. Heuts and I. Moerdijk. Simplicial and dendroidal homotopy theory. Springer, Cham (2022).
- J. Lurie. Higher algebras. Available online.
On the theory of operadic Koszul duality
- L. Brantner and R. Campos and J. Nuiten. PD Operads and Explicit Partition Lie Algebras. Preprint. Available at arXiv (2021).
- J. Lurie. Higher algebras. Available online.
On applications
- O. A. Camarena. The mod 2 homology of free spectral Lie algebras. Trans. Am. Math. Soc. 373, No. 9, 6301--6319 (2020).
- G. Heuts. Koszul duality and a conjecture of Francis-Gaitsgory. Preprint. Available at arXiv (2024).
- G. Heuts. Lie algebra models for unstable homotopy theory. Handbook of homotopy theory. CRC Press, Boca Raton, FL. 657--698 (2020)
- N. J. Kuhn. Localization of André-Quillen-Goodwillie towers, and the periodic homology of infinite loopspaces. Adv. Math. 201, No. 2, 318--378 (2006).
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).
- M. Shulmann. Set theory for category theory. Available at arXiv.
On coalgebras
- M. Anel. Cofree coalgebras over operads and representative functions. Available at arXiv.
- G. Heuts. Koszul duality and a conjecture of Francis-Gaitsgory. Available at arXiv.
- M. Péroux. The coalgebraic enrichment of algebras in higher categories. J. Pure Appl. Algebra 226, No. 3, Article ID 106849, 11 p. (2022)
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.