nLab simplicial homotopy theory

Idea

Simplicial homotopy theory is the study of homotopy theory by means of simplicial sets, but also the study of those properties of simplicial sets detectable by means of techniques adapted from topological homotopy theory.

A key tool is the classical model structure on simplicial sets.

Related entries

• simplex category, simplicial identities,

• simplicial object,

  • simplicial set, simplicial group, simplicial topological space,

• nerve

• singular simplicial complex

• simplicial classifying space, simplicial loop space

• Kan fibration, Kan complex,

  • anodyne morphism, left fibration, minimal fibration

• simplicial homotopy

• simplicial homotopy group

• geometric realization

  • geometric realization of categories

  • geometric realization of simplicial topological spaces

• weak homotopy equivalence

• model structure on simplicial sets

• homotopy hypothesis-theorem

References

(See also the references at homotopy theory, algebraic topology and $(\infty,1)$-category theory.)

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

• 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)

• Garth Warner: Categorical Homotopy Theory, EPrint Collection, University of Washington (2012) [hdl:1773/19589, pdf, pdf]

• Zhen Lin Low, Notes on homotopical algebra

• Emily Riehl, Categorical Homotopy Theory

• Jacob Lurie, Higher Topos Theory

• Jacob Lurie, Kerodon