Homotopical Algebra
- General
The main task of this course is to give an introduction to the methods of homotopical algebra. This is a 6 credits advanced course of 42 hour lectures. See 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 generalization of homological algebra. A special regard to algebraic geometry will provide some applications to motivic homotopy and algebraic K-Theory.
- Prerequisites
We assume known the basic notions from algebraic topology & homological algebra. The students should be familiar with fundamental group, singular homology/cellular homology, CW-complexes and 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 are not available as yet. However, a good source is given by Jardine's Lectures on Homotopy Theory.
- 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
- A^{1}-homotopy & homotopy invariance
- Exams
Some homeworks 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 SIFA form online here to get your name listed for the exam (but the page is in italian so you might need help). Those who for some reason wish to postpone the examination procedure can make a motivated agreement (not by email) with me for a later date if there are good reasons for that.
- References
Extras
Homotopy Theory
- G. Maltsiniotis: La théorie de l'homotopie de Grothendieck Astérisque, 301 (2005) & Pursuing Stacks dedicated web page.
- P.S. Hirschhorn: Model categories and their localizations Math Surveys & Monographs Vol. 99 AMS, 2003.
- D. Dagger: Universal homotopy theories Adv. Math. 164 (2001), no. 1, 144-176
- P. G. Goerss & J. F. Jardine: Simplicial homotopy theory Progress in Mathematics Vol. 174 Birkhauser Verlag, 1999.
- W.G. Dwyer & J. Spalinski: Homotopy theories and model categories in Handbook of Algebraic Topology I.M. James (ed.) North-Holland, 1995
- 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 to appear in the series Cambridge studies in advanced mathematics, Cambridge University Press.
- 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
- 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: A^{1}-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.