nLab homotopy theory and algebraic topology -- references

The following lists basic references on homotopy theory, algebraic topology and some $(\infty,1)$-category theory and homotopy type theory, but see these entries for more pointers.

Pre-history

Historical article at the origin of all these subjects:

• Henri Poincaré, Analysis Situs, Journal de l’École Polytechnique. (2). 1: 1–123 (1895) (gallica:12148/bpt6k4337198/f7, Engl: pdf, pdf)

On early developments from there, such as the eventual understanding of the notion of higher homotopy groups:

• Peter Hilton, Subjective History of Homology and Homotopy Theory, Mathematics Magazine 61 5 (1988) 282-291 $[$doi:10.2307/2689545$]$

Topological homotopy theory

Textbook accounts of homotopy theory of topological spaces (i.e. via “point-set topology”):

• Peter J. Hilton, An introduction to homotopy theory, Cambridge University Press 1953 (doi:10.1017/CBO9780511666278)

• Sze-Tsen Hu, Homotopy Theory, Academic Press 1959 (pdf)

• Robert E. Mosher, Martin C. Tangora, Cohomology operations and applications in homotopy theory, Harper & Row, 1968, reprinted by Dover 2008 GoogleBooks

• Tammo tom Dieck, Klaus Heiner Kamps, Dieter Puppe, Homotopietheorie, Lecture Notes in Mathematics 157 Springer 1970 (doi:10.1007/BFb0059721)

• Brayton Gray, Homotopy Theory: An Introduction to Algebraic Topology, Academic Press (1975) $[$978-0-12-296050-5, pdf$]$

• George W. Whitehead, Elements of Homotopy Theory, Springer 1978 (doi:10.1007/978-1-4612-6318-0)

• Ioan Mackenzie James, General Topology and Homotopy Theory, Springer 1984 (doi:10.1007/978-1-4613-8283-6)

• Renzo A. Piccinini, Lectures on Homotopy Theory, Mathematics Studies 171, North Holland 1992 (ISBN:978-0-444-89238-6)

• Glen Bredon, Chapter VII of: Topology and Geometry, Graduate texts in mathematics 139, Springer 1993 (doi:10.1007/978-1-4757-6848-0, pdf)

• Hans-Joachim Baues, Homotopy types, in Ioan Mackenzie James (ed.) Handbook of Algebraic Topology, North Holland, 1995 (ISBN:9780080532981, doi:10.1016/B978-0-444-81779-2.X5000-7)

• Nicolas Bourbaki, Topologie Algébrique, Chapitres 1 à 4, Springer (1998, 2016) [ISBN 978-3-662-49361-8, doi:10.1007/978-3-662-49361-8]

• Marcelo Aguilar, Samuel Gitler, Carlos Prieto, Algebraic topology from a homotopical viewpoint, Springer (2008) (doi:10.1007/b97586)

• Jeffrey Strom, Modern classical homotopy theory, Graduate Studies in Mathematics 127, American Mathematical Society (2011) $[$doi:10.1090/gsm/127$]$

• Martin Arkowitz, Introduction to Homotopy Theory, Springer (2011) $[$doi:10.1007/978-1-4419-7329-0$]$

• Dai Tamaki, Fiber Bundles and Homotopy, World Scientific (2021) $[$doi:10.1142/12308$]$

  (motivated from classifying spaces for principal bundles/fiber bundles)

Algebraic topology

On algebraic topology:

Textbooks:

• Samuel Eilenberg, Norman Steenrod, Foundations of Algebraic Topology, Princeton University Press 1952 (pdf, ISBN:9780691653297)

• Roger Godement, Topologie algébrique et theorie des faisceaux, Actualités Sci. Ind. 1252, Hermann, Paris (1958) $[$webpage, pdf$]$

• Edwin Spanier, Algebraic topology, McGraw Hill (1966), Springer (1982) (doi:10.1007/978-1-4684-9322-1)

• William S. Massey, Algebraic Topology: An Introduction, Harcourt Brace & World 1967, reprinted in: Graduate Texts in Mathematics, Springer 1977 (ISBN:978-0-387-90271-5)

• C. R. F. Maunder, Algebraic Topology, Cambridge University Press, Cambridge (1970, 1980) $[$pdf$]$

• Robert Switzer, Algebraic Topology - Homotopy and Homology, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975 (doi:10.1007/978-3-642-61923-6)

• Raoul Bott, Loring Tu, Differential Forms in Algebraic Topology, Graduate Texts in Mathematics 82, Springer (1982) $[$doi:10.1007/978-1-4757-3951-0]

  (with focus on differential forms, differential topology)

• James Munkres, Elements of Algebraic Topology, Addison-Wesley (1984) $[$pdf]

• Joseph J. Rotman, An Introduction to Algebraic Topology, Graduate Texts in Mathematics 119 (1988) $[$doi:10.1007/978-1-4612-4576-6]

• Glen Bredon, Topology and Geometry, Graduate texts in mathematics 139, Springer 1993 (doi:10.1007/978-1-4757-6848-0, pdf)

• Albrecht Dold, Lectures on Algebraic Topology, Springer 1995 (doi:10.1007/978-3-642-67821-9, pdf)

• William Fulton, Algebraic Topology – A First Course, Graduate Texts in Mathematics 153, Springer (1995) [doi:10.1007/978-1-4612-4180-5]

• Peter May, A concise course in algebraic topology, University of Chicago Press 1999 (ISBN: 9780226511832, pdf)

• Tammo tom Dieck, Topologie, De Gruyter 2000 (doi:10.1515/9783110802542)

• Peter May, Kate Ponto, More concise algebraic topology, University of Chicago Press (2012) (ISBN:9780226511795, pdf)

• Dai Tamaki, Akira Kono, Generalized Cohomology, Translations of Mathematical Monographs, American Mathematical Society, 2006 (ISBN: 978-0-8218-3514-2)

• Tammo tom Dieck, Algebraic topology, European Mathematical Society, Zürich (2008) (doi:10.4171/048, pdf)

• Allen Hatcher, Algebraic Topology, Cambridge University Press (2002) $[$ISBN:9780521795401, webpage$]$

On constructive methods (constructive algebraic topology):

• Julio Rubio, Francis Sergeraert, Constructive Algebraic Topology, Bulletin des Sciences Mathématiques

  126 5 (2002) 389-412 [doi:10.1016/S0007-4497(02)01119-3, arXiv:math/0111243]

Lecture notes:

• Michael Hopkins (notes by Akhil Mathew), algebraic topology – Lectures (pdf)

• Friedhelm Waldhausen, Algebraische Topologie I (pdf) , II (pdf), III (pdf) (web)

• James F. Davis and Paul Kirk, Lecture notes in algebraic topology (pdf)

• Gereon Quick, Advanced algebraic topology, 2014

Survey of various subjects in algebraic topology:

• Ioan Mackenzie James, Handbook of Algebraic Topology 1995

Survey with relation to differential topology:

• Sergei Novikov, Topology I – General survey, in: Encyclopedia of Mathematical Sciences Vol. 12, Springer 1986 (doi:10.1007/978-3-662-10579-5, pdf)

• Jean Dieudonné, A History of Algebraic and Differential Topology, 1900 - 1960, Modern Birkhäuser Classics 2009 (ISBN:978-0-8176-4907-4)

With focus on ordinary homology, ordinary cohomology and abelian sheaf cohomology:

• Jean Gallier, Jocelyn Quaintance, Homology, Cohomology, and Sheaf Cohomology for Algebraic Topology, Algebraic Geometry, and Differential Geometry, World Scientific (2022) [doi:10.1142/12495, webpage]

Some interactive 3D demos:

• Neil Strickland, Interactive pages for Algebraic Topology, web site

Further pointers:

• a thread on AlgTop literature at MathOverflow

Abstract homotopy theory

On localization at weak equivalences to homotopy categories:

• Edgar Brown, Abstract homotopy theory, Trans. AMS 119 no. 1 (1965) (doi:10.1090/S0002-9947-1965-0182970-6)

On localization via calculus of fractions:

• Pierre Gabriel, Michel Zisman, Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35. Springer, New York (1967)

On localization via model category-theory:

• Daniel Quillen, Homotopical algebra, Lecture Notes in Mathematics 43, Berlin, New York, 1967

• Mark Hovey, Model Categories, Mathematical Surveys and Monographs, Volume 63, AMS (1999) (ISBN:978-0-8218-4361-1, doi:10.1090/surv/063, pdf, Google books)

• Philip Hirschhorn, Model Categories and Their Localizations, AMS Math. Survey and Monographs Vol 99 (2002) (ISBN:978-0-8218-4917-0, pdf toc, pdf)

• William G. Dwyer, Philip S. Hirschhorn, Daniel M. Kan, Jeffrey H. Smith, Homotopy Limit Functors on Model Categories and Homotopical Categories, Mathematical Surveys and Monographs 113 (2004) $[$ISBN: 978-1-4704-1340-8, pdf$]$

On localization (especially of categories of simplicial sheaves/simplicial presheaves) via categories of fibrant objects:

• Kenneth S. Brown, Abstract Homotopy Theory and Generalized Sheaf Cohomology, Transactions of the American Mathematical Society 186 (1973) 419-458 $[$jstor:1996573$]$.

See also:

• Klaus Heiner Kamps, Tim Porter, Abstract Homotopy and Simple Homotopy Theory, World Scientific 1997 (doi:10.1142/2215, GoogleBooks)

• Haynes Miller (ed.), Handbook of Homotopy Theory, 2019

Lecture notes:

• William Dwyer, Homotopy theory and classifying spaces, Copenhagen, June 2008 (pdf, pdf)

• Jesper Michael Møller, Homotopy theory for beginners, 2015 (pdf, pdf)

• Urs Schreiber, Introduction to Homotopy Theory (2016)

• Yuri Ximenes Martins, Introduction to Abstract Homotopy Theory (arXiv:2008.05302)

Introduction, from category theory to (mostly abstract, simplicial) homotopy theory:

• Emily Riehl, Categorical Homotopy Theory, Cambridge University Press, 2014 (pdf, doi:10.1017/CBO9781107261457)

• Birgit Richter, From categories to homotopy theory, Cambridge Studies in Advanced Mathematics 188, Cambridge University Press 2020 (doi:10.1017/9781108855891, book webpage, pdf)

• Urs Schreiber, geometry of physics – categories and toposes

See also:

• William Dwyer, Philip Hirschhorn, Daniel Kan, Jeff Smith, Homotopy Limit Functors on Model Categories and Homotopical Categories, volume 113 of Mathematical Surveys and Monographs, American Mathematical Society (2004) (there exists this pdf copy of what seems to be a preliminary version of this book)

• Zhen Lin Low, Notes on homotopical algebra, 2015

• Brian Munson, Ismar Volic, Cubical homotopy theory, Cambridge University Press, 2015 (pdf, doi:10.1017/CBO9781139343329)

  (with emphasis on cubical objects such as in n-excisive functors and Goodwillie calculus)

Simplicial homotopy theory

On simplicial homotopy theory:

• Peter May, Simplicial objects in algebraic topology, University of Chicago Press 1967 (ISBN:9780226511818, djvu, pdf)

• Edward B. Curtis, Simplicial homotopy theory, Advances in Mathematics 6 (1971) 107–209 (doi:10.1016/0001-8708(71)90015-6, MR279808)

• Paul Goerss, J. F. Jardine, Section V.4 of: Simplicial homotopy theory, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) (doi:10.1007/978-3-0346-0189-4, webpage)

• André Joyal, Myles Tierney Notes on simplicial homotopy theory, Lecture at Advanced Course on Simplicial Methods in Higher Categories, CRM 2008 (pdf)

• André Joyal, Myles Tierney, An introduction to simplicial homotopy theory, 2009 (web, pdf)

• Paul Goerss, Kirsten Schemmerhorn, Model categories and simplicial methods, Notes from lectures given at the University of Chicago, August 2004, in: Interactions between Homotopy Theory and Algebra, Contemporary Mathematics 436, AMS 2007(arXiv:math.AT/0609537, doi:10.1090/conm/436)

• Francis Sergeraert, Introduction to Combinatorial Homotopy Theory, 2008 (pdf, pdf)

Basic $(\infty,1)$-category theory

On (∞,1)-category theory and (∞,1)-topos theory:

• André Joyal, The theory of quasicategories and its applications lectures at Advanced Course on Simplicial Methods in Higher Categories, CRM 2008 (pdf, pdf)

• André Joyal, Notes on Logoi, 2008 (pdf, pdf)

• Jacob Lurie, Higher Topos Theory

• Denis-Charles Cisinski, Higher category theory and homotopical algebra (pdf)

Basic homotopy type theory

On synthetic homotopy theory in homotopy type theory:

Exposition:

• Dan Licata: Homotopy theory in type theory (2013) $[$pdf slides, pdf, blog entry 1, blog entry 2$]$

• Mike Shulman, The logic of space, in: Gabriel Catren, Mathieu Anel (eds.), New Spaces for Mathematics and Physics, Cambridge University Press (2021) 322-404 $[$arXiv:1703.03007, doi:10.1017/9781108854429.009$]$

Textbook accounts:

• Univalent Foundations Project: Homotopy Type Theory – Univalent Foundations of Mathematics (2013) (webpage, pdf)

• Egbert Rijke, Introduction to Homotopy Type Theory (2019) (web, pdf, GitHub)

For more see also at homotopy theory formalized in homotopy type theory.

Outlook

Indications of open questions and possible future directions in algebraic topology and (stable) homotopy theory:

• Mark Hovey, Algebraic Topology Problem List

• Tyler Lawson, The future, Talbot lectures 2013 (pdf)

• Problems in homotopy theory (wiki)

More regarding the sociology of the field (such as its folklore results):

• Clark Barwick, The future of homotopy theory, 2017 (pdf, pdf)