nLab classical model structure on simplicial sets -- references

Classical model structure on simplicial sets

On the classical model structure on simplicial sets:

The original proof is due to

• Daniel Quillen, Section II.3 of: Axiomatic homotopy theory in: Homotopical Algebra, Lecture Notes in Mathematics 43, Springer 1967(doi:10.1007/BFb0097438)

This proof is purely combinatorial (i.e. does not pass through geometric realization of simplicial sets as topological spaces): Quillen uses the theory of minimal Kan fibrations, the fact that the latter are fiber bundles, as well as the fact that the simplicial classifying space of a simplicial group is a Kan complex.

Other proofs are were given in:

• Sergei Gelfand, Yuri Manin, Sections V.1-2 of: Methods of homological algebra, transl. from the 1988 Russian (Nauka Publ.) original, Springer 1996. xviii+372 pp. 2nd corrected ed. 2002 (doi:10.1007/978-3-662-12492-5)

• Paul Goerss, J. F. Jardine, Section I.11 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, 2005 (web, pdf)

A proof (in fact two variants of it) using the Kan fibrant replacement $Ex^\infty$ functor is given (in the context of_Cisinski model structure) in:

• Denis-Charles Cisinski, Section 2 of: Les préfaisceaux comme type d'homotopie, Astérisque, Volume 308, Soc. Math. France (2006), 392 pages (pdf)

The crucial step is the proof that the fibrations are precisely the Kan fibrations (and also to prove all the good properties of $Ex^\infty$ without using topological spaces); for two different proofs of this fact using $Ex^\infty$, see Prop. 2.1.41 as well as Scholium 2.3.21 for an alternative). For the rest, everything was already in the book of Gabriel and Zisman, for instance.

Another approach using $Ex^\infty$ is:

• Sean Moss, Another approach to the Kan-Quillen model structure, Journal of Homotopy and Related Structures volume 15, pages 143–165 (2020) (arXiv:1506.04887, doi:10.1007/s40062-019-00247-y)

A proof of the model structure not relying on the classical model structure on topological spaces nor on explicit models for Kan fibrant replacement is in

• Christian Sattler, The Equivalence Extension Property and Model Structures (arXiv:1704.06911)

Proofs valid in constructive mathematics are given in:

• Simon Henry, A constructive account of the Kan-Quillen model structure and of Kan’s Ex∞ functor, arXiv:1905.06160.

• Nicola Gambino, Simon Henry, Christian Sattler, Karol Szumiło, The effective model structure and ∞-groupoid objects, arXiv:2102.06146.

As a categorical semantics for homotopy type theory, the model structure on simplicial sets is considered in

• Chris Kapulkin, Peter LeFanu Lumsdaine, Vladimir Voevodsky, (arXiv:1203.2553)

• Chris Kapulkin, Peter LeFanu Lumsdaine, The Simplicial Model of Univalent Foundations (after Voevodsky), Journal of the European Mathematical Society (arXiv:1211.2851,doi:10.4171/jems/1050)