Abstract
One may think of homotopical algebra as a tool for computing and systematically studying obstructions to the resolution of (not necessarily linear) problems. Since most of the problems that occur in physics and mathematics carry obstructions, one needs tools to study these and give an elegant presentation of the physicists’ ideas (who often invented some of these techniques for their own safety). In this chapter, we present the main tools that allow the computation of obstructions, by giving a concrete flavor to the idea of general obstruction theory. We start by discussing localizations and derived functors, and then discuss the setting of model categories. We also discuss the theory of simplicial sets, as concrete incarnations of higher groupoids and ∞-categories. We then study derived categories and derived operations on sheaves, and finish by a presentation of a model for the theory of higher categories, and of their use in categorical logic.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ara, D.: The groupoidal analogue Theta˜ to Joyal’s category Theta is a test category. arXiv e-prints (2010). arXiv:1012.4319
Ara, D.: Higher quasi-categories vs higher Rezk spaces. arXiv e-prints (2012). arXiv:1206.4354
Baez, J.C., Dolan, J.: Categorification. arXiv Mathematics e-prints (1998). arXiv:math/9802029
Bergner, J.E.: A survey of (∞,1)-categories. arXiv Mathematics e-prints (2006). arXiv:math/0610239
Bousfield, A.K., Kan, D.M.: Homotopy Limits, Completions and Localizations. Lecture Notes in Mathematics, vol. 304, p. 348. Springer, Berlin (1972)
Berger, C., Moerdijk, I.: Axiomatic homotopy theory for operads. Comment. Math. Helv. 78(4), 805–831 (2003). doi:10.1007/s00014-003-0772-y
Bergner, J.E., Rezk, C.: Comparison of models for (∞,n)-categories, I. arXiv e-prints (2012). arXiv:1204.2013
Barwick, C., Schommer-Pries, C.: On the unicity of the homotopy theory of higher categories. arXiv e-prints (2011). arXiv:1112.0040
Cisinski, D.-C., Déglise, F.: Local and stable homological algebra in Grothendieck Abelian categories. Homol. Homotopy Appl. 11(1), 219–260 (2009)
Cisinski, D.-C.: Catégories dérivables. Bull. Soc. Math. Fr. 138(3), 317–393 (2010)
Dwyer, W.G., Hirschhorn, P.S., Kan, D.M., Smith, J.H.: Homotopy Limit Functors on Model Categories and Homotopical Categories. Mathematical Surveys and Monographs, vol. 113, p. 181. Am. Math. Soc., Providence (2004). ISBN 0-8218-3703-6
Dwyer, W.G., Kan, D.M.: Simplicial localizations of categories. J. Pure Appl. Algebra 17(3), 267–284 (1980). doi:10.1016/0022-4049(80)90049-3
Dwyer, W.G., Spaliński, J.: Homotopy theories and model categories. In: Handbook of Algebraic Topology, pp. 73–126. North-Holland, Amsterdam (1995). doi:10.1016/B978-044481779-2/50003-1
Fresse, B.: Modules over Operads and Functors. Lecture Notes in Mathematics, vol. 1967, p. 308. Springer, Berlin (2009). doi:10.1007/978-3-540-89056-0. ISBN 978-3-540-89055-3
Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory. Progress in Mathematics, vol. 174, p. 510. Birkhäuser, Basel (1999). ISBN 3-7643-6064-X
Gabriel, P., Zisman, M.: Calculus of Fractions and Homotopy Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 35, p. 168. Springer, New York (1967)
Hartshorne, R.: Residues and Duality. Springer, Berlin (1970)
Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 52, p. 496. Springer, New York (1977). ISBN 0-387-90244-9
Hovey, M.: Model Categories. Mathematical Surveys and Monographs, vol. 63, p. 209. Am. Math. Soc., Providence (1999). ISBN 0-8218-1359-5
Joyal, A.: Discs, duality and theta-categories (2007, preprint)
Joyal, A.: Notes on quasi-categories (2011, preprint)
Keller, B.: A ∞-algebras, modules and functor categories. In: Trends in Representation Theory of Algebras and Related Topics. Contemp. Math., vol. 406, pp. 67–93. Am. Math. Soc., Providence (2006)
Kahn, B., Maltsiniotis, G.: Structures de dérivabilité. Adv. Math. 218(4), 1286–1318 (2008)
Kashiwara, M., Schapira, P.: Categories and Sheaves. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 332, p. 497. Springer, Berlin (2006). ISBN 978-3-540-27949-5; 3-540-27949-0
Lurie, J.: On the classification of topological field theories. arXiv e-prints (2009). arXiv:0905.0465
Lurie, J.: Derived algebraic geometry. Doctoral thesis (2009, preprint)
Lurie, J.: Higher algebra (2009, preprint)
Lurie, J.: Higher Topos Theory. Annals of Mathematics Studies, vol. 170, p. 925. Princeton University Press, Princeton (2009). ISBN 978-0-691-14049-0; 0-691- 14049-9
Loday, J.-L., Vallette, B.: Algebraic Operads. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 346. Springer, Heidelberg (2012). ISBN 978-3-642-30361-6. xxiv+634 pp.
Merkulov, S.A.: Wheeled pro(p)file of Batalin-Vilkovisky formalism. Commun. Math. Phys. 295, 585–638 (2010). doi:10.1007/s00220-010-0987-x. arXiv:0804.2481
Quillen, D.G.: Homotopical Algebra. Lecture Notes in Mathematics, vol. 43, p. 156. Springer, Berlin (1967)
Rezk, C.: A cartesian presentation of weak n-categories. arXiv e-prints (2009). arXiv:0901.3602
Simpson, C.T.: Homotopy theory of higher categories. arXiv e-prints (2010). arXiv:1001.4071
Tamsamani, Z.: On non-strict notions of n-category and n-groupoid via multisimplicial sets (1995). arXiv:math.AG/9512006
Tamarkin, D.E.: Another proof of M. Kontsevich formality theorem. arXiv Mathematics e-prints (1998). arXiv:math/9803025
Toen, B.: From homotopical algebra to homotopical algebraic geometry: lectures in Essen (2002). http://www.math.univ-toulouse.fr/~toen/essen.pdf
Toen, B.: Towards an axiomatization of the theory of higher categories. arXiv Mathematics e-prints (2004). arXiv:math/0409598
Toen, B., Vezzosi, G.: Homotopical algebraic geometry I: topos theory. arXiv Mathematics e-prints (2002). arXiv:math/0207028
Toën, B., Vezzosi, G.: Homotopical Algebraic Geometry. II. Geometric Stacks and Applications. Mem. Am. Math. Soc. 193(902):x+224 (2008)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Paugam, F. (2014). Homotopical Algebra. In: Towards the Mathematics of Quantum Field Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol 59. Springer, Cham. https://doi.org/10.1007/978-3-319-04564-1_9
Download citation
DOI: https://doi.org/10.1007/978-3-319-04564-1_9
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-04563-4
Online ISBN: 978-3-319-04564-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)