Model structure on Simplicial Sets without using topological spaces

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.

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.

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 and only if one of the following equivalent conditions is satisfied:

• f has the right homotopy lifting property with respect to Sd^i ∂Δ^n → Sd^i Δ^n (allowing subdivisions for homotopies also).
• Ex^∞(f) has the right homotopy lifting property with respect to ∂Δ^n→Δ^n.
• Ex^∞(f) is a simplicial homotopy equivalence.
• Ex^∞(f) induces an isomorphism on π_0 and all homotopy groups for any choice of basepoints.
• Ex^∞(f) induces isomorphisms on simplicial homotopy groups.
• Hom(f, A) is a simplicial homotopy equivalence for every Kan complex A.
• The morphism f is a composition of a trivial cofibration and a trivial fibration, both of which are defined using lifting properties.
• Applying the category of elements functor produces a Thomason weak equivalence of categories. The class of Thomason weak equivalences forms the smallest basic localizer, i.e., the smallest class of functors between small categories that contains identities, is closed under retracts and the 2-out-of-3 property, contains all functors A→1 for which the category A has a terminal object, and is locally determined: if u:A→B and w:B→C are functors, with v=w∘u:A→C, and for any $c∈C$ the induced functor of comma categories v/c→w/c is a Thomason weak equivalence, then so is u.

Gelfand and Manin's Methods of Homological Algebra contains a sketchy construction of the standard model structure on simplicial sets without referring to topological spaces.