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I have split off an entry classical model structure on simplicial sets from “model structure on simplicial set”. This entry should eventually contain detailed, self-contained and polished discussion of the definition, verification and key properties of the standard Kan-Quillen model structure.
So far I have inserted fair bit of background material regarding (minimal) fibrations and geometric realizations, essentially the material in chapter 1 of Goerss-Jardine. A bunch of little proofs are spelled out, but not yet the more laborious ones. Discussion of the verification of the axioms is not yet in the entry, but the key parts of the Quillen equivalence to $Top_{Quillen}$ are (modulo relying on previous lemmas that don’t have proofs spelled out yet).
The somewhat random list of properties of $sSet_{Quillen}$ that used to be sitting at “model structure on simplicial sets” I have copied over to a section “Basic properties”, just for completenes, but this now needs re-organization to give decent logical flow.
For the moment I have to leave it at that, need to take care of something else now for a little bit.
added precise relevant section number and also full publication data for
and added links to the electronic version of
Just so you know, the “more notes pdf” link is broken.
So I cleaned up this mess and have hosted the files now on the nLab server:
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)
It looks like at CRM in 2008 they were lacking somebody with a little familiarity with the internet: The authors don’t date their pdf-s and don’t cite the event they are contributions to; while the organizers either don’t host the documents, or where they do, don’t ensure that the hosting is stable.
Thanks! I had no idea where to start on that one.
Added:
Constructive proofs can be found 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.
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