Model structure on Simplicial Sets without using topological spaces - MathOverflow

Model structure on Simplicial Sets without using topological spaces

2010-11-14T23:54:34Z

The category of simplicial sets has a standard model structure, where the weak equivalences are those maps whose geometric realization is a weak homotopy equivalence, the cofibrations are monomorphisms, and the fibrations are Kan fibrations.

Simplicial sets are combinatorial objects, so morally their model structure should not be dependent on topological spaces. Are there any approaches to this model structure which do not use the geometric realization functor, and do not use topological spaces?

Answer by Dan Ramras for Model structure on Simplicial Sets without using topological spaces

2010-11-15T00:03:03Z

Yes. The only one I know of is in a book by Joyal and Tierny. I heard some time ago that the book was going to be published, but I don't know if that has happened. There's a version on the Hopf topology archive:

http://hopf.math.purdue.edu/cgi-bin/generate?/Joyal-Tierney/JT-chap-01

If you look at the first page, they state what you're looking for as their main goal.

If anyone knows of a more recent version, maybe with more chapters, let us know!

Answer by Harry Gindi for Model structure on Simplicial Sets without using topological spaces

2010-11-15T00:07:39Z

Denis-Charles Cisinski has a beautiful book called Les Préfaisceaux commes modèles des Types d'Homotopie, which gives a very very powerful framework for building model structures on presheaf categories (and more generally Grothendieck toposes), and after building up this framework, the model structure for simplicial sets drops out literally for free.

Here's a link to it from his website: link.

He also proves some nontrivial conjectures of Grothendieck that are important for derivator theory, among other things. Rick Jardine published a summary paper of this book, which is also worth reading.

Note: The framework is built up entirely in chapter 1, so even if you don't want to read the whole book, the first chapter is what you need.

Answer by Dmitri Pavlov for Model structure on Simplicial Sets without using topological spaces

2010-11-15T11:50:56Z

There are many ways to define weak equivalences of simplicial sets without referring to topological spaces.

A morphism f is a weak equivalence of simplicial sets if one of the following equivalent conditions is satisfied:

Ex^∞(f) is a simplicial homotopy equivalence.
Ex^∞(f) induces isomorphism on simplicial homotopy groups.
Hom(f, A) is an isomorphism for every Kan complex (fibrant simplicial set) A.
The morphism f is a composition of a trivial cofibration and a trivial fibration.

Note that fibrations and trivial fibrations, as well as cofibrations and trivial cofibrations can be defined using left and right lifting properties.

A book on homological algebra by Gelfand and Manin contains a sketchy construction of the standard model structure on simplicial sets without referring to topological spaces.

Answer by Denis-Charles Cisinski for Model structure on Simplicial Sets without using topological spaces

2010-11-15T12:04:34Z

Quillen's original proof (in Homotopical Algebra, LNM 43, Springer, 1967) is purely combinatorial (i.e. does not use topological spaces): he uses the theory of minimal Kan fibrations, the fact that the latter are fiber bundles, as well as the fact that the classifying space of a simplicial group is a Kan complex. This proof has been rewritten several times in the literature: at the end of

S.I. Gelfand and Yu. I. Manin, Methods of Homological Algebra, Springer, 1996

as well as in

A. Joyal and M. Tierney An introduction to simplicial homotopy theory

(I like Joyal and Tierney's reformulation a lot). However, Quillen wrote in his seminal Lecture Notes that he knew another proof of the existence of the model structure on simplicial sets, using Kan's $Ex^\infty$ functor (but does not give any more hints).

A proof (in fact two variants of it) using Kan's $Ex^\infty$ functor is given in my Astérisque 308: the fun part is not that much about the existence of model structure, but to prove 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.

Finally, I would even add that, in Quillen's original paper, the model structure on topological spaces in obtained by transfer from the model structure on simplicial sets. And that is indeed a rather natural way to proceed.