Homotopical Algebra - academic year 2023/24

• General The main task of this course is to give an introduction to the methods of homotopical algebra and ∞-categories. This is a 6 credits advanced course of 42 hour lectures. There is an official Ariel Portal homepage Homotopical Algebra. See also Algant local page for schedules, etc.


• Course (6 credits)
  
• Description Homotopical algebra studies homotopies appearing in several mathematical spots: from topology to algebraic geometry and more. Notably, homotopical algebra can be regarded as a generalisation of homological algebra.


• Prerequisites We assume known the basic notions from category theory, algebraic topology & homological algebra. The students should be familiar with the fundamental group, singular homology, cellular homology, CW-complexes and the related basic computations. For example, see A. Hatcher Algebraic Topology Cambridge Univ. Press, 2002. In a few lectures we will treat some topology using J. P. May A concise course in Algebraic Topology Chicago Lectures in Mathematics 1999. For homological algebra we just assume known the basics on chain complexes as explained in the first chapter of C. Weibel's book An introduction to Homological Algebra Cambridge Univ. Press, 1994.


• Notes Course notes can be (partially) extracted from Jardine's Lectures on Homotopy Theory. You can get homework through the Ariel Portal homepage Homotopical Algebra.


• Program (key words & outline)
• homotopy & homology
• weak equivalences & quasi isomorphisms
• fibrations & cofibrations
• model categories & homotopy categories
• Quillen functors, derived functors & equivalences
• simplicial homotopy & geometric realisation
• nerve of a category & ∞-categories
• universal homotopy & motivic homotopy


• Extras Higher algebra is a natural outcome of homotopical algebra. Moreover, motivic homotopy & homology is a significant application of homotopical algebra to algebraic geometry that is hinted for the interested students having some knowledge of the fundamentals of the theory of schemes. Prerequisites on schemes are contained in M. Levine's Elementary Algebraic Geometry or any other introductory book on the subject.


• Exams Homework will be assigned during the lectures. Preferably, the solutions shall be provided within the end of the course. Next a seminar on your favorite subject will be assigned according to the themes hinted in class. Please check here the dates for the current academic year. You need to fill the form online here to get your name listed for the exam. Those who for some reason wish to postpone the examination procedure can make a motivated agreement (by email) with me for a later date.


References
• M. Hovey: Model Categories Math Surveys & Monographs Vol. 63 AMS 1999.
• W.G. Dwyer & J. Spalinski: Homotopy theories and model categories in Handbook of Algebraic Topology I.M. James (ed.) North-Holland, 1995.
• J. Lurie: Kerodon is an online resource for homotopy-coherent higher mathematics.


Extras
Homotopy Theory
• P. G. Goerss & J. F. Jardine: Simplicial homotopy theory Progress in Mathematics Vol. 174 Birkhauser Verlag, 1999.
• P.S. Hirschhorn: Model categories and their localizations Math Surveys & Monographs Vol. 99 AMS, 2003.
• D. Dugger: Universal homotopy theories Adv. Math. 164 (2001), no. 1, 144-176.
• G. Maltsiniotis: La théorie de l'homotopie de Grothendieck Astérisque, 301 (2005) & Pursuing Stacks dedicated web page.
• D. Quillen: "Homotopical Algebra" Springer LNM 43, 1967
• P. Gabriel & M. Zisman: "Calculus of fractions and homotopy theory" Springer, 1967
Higher Algebra
• D.-C. Cisinski: Higher categories and homotopical algebra Cambridge studies in advanced mathematics, vol. 180, Cambridge University Press, 2019. xviii+430 pp. + Errata
• J. Lurie: Higher Topos Theory Annals of Mathematics Studies, vol. 170, Princeton University Press, Princeton, NJ, 2009.
• J. Lurie: Higher Algebra Preliminary Version, 2017.
Motivic homotopy & homology
• B.I. Dundas, M. Levine, P.A. Ostvaer & V. Voevodsky: Motivic Homotopy Theory Lectures at a Summer School in Nordfjordeid (Norway, August 2002) Springer Universitext, 2007
• F. Morel & V. Voevodsky: A¹-homotopy theory of schemes Publications Mathématiques de l'IHÉS Vol. 90 (1999), p. 45-143
• C. Mazza, V. Voevodsky & C. Weibel: Lecture Notes on Motivic Cohomology Clay Mathematics Monographs, Vol. 2, 2006 (Lectures given by V. Voevodsky at IAS in 1999-2000)
K-Theory
• C. Weibel: An introduction to algebraic K-theory Graduate Studies in Math. vol. 145, AMS, 2013 & also available on line at C. Weibel's home page
• E.M. Friedlander & D.R. Grayson (ed.) Handbook of K-theory Vol. 1, 2 Springer-Verlag, 2005
• W. Fulton & S. Lang: "Riemann-Roch Algebra" Springer Grundlehren der mathematischen Wissenschaften, Vol. 277, 1985.