There are several people here much more qualified to speak about that, so I shall just give you some pointers now. One of the questions Grothendieck tried to answer when writing "Pursuing Stacks" was — I don't know how he put it, though — "what are the properties of the simplicial category which make it so useful in homotopy theory?" That is where the theory of *test categories* stems from. As Georges Maltsiniotis puts it: "Le slogan de Grothendieck est que toute catégorie test est aussi “bonne” que celle des ensembles simpliciaux pour 
“faire de l’homotopie”." Which means "Grothendieck's motto is that any test category is as "good" as the category of simplicial sets to make homotopy theory." The theory was further developed by Denis-Charles Cisinski. The two books to read on this subject are:

Maltsiniotis's "La Théorie de l'homotopie de Grothendieck":
http://www.math.jussieu.fr/~maltsin/ps/prstnew.pdf

and

Cisinski's (augmented version of his) thesis "Les Préfaisceaux comme modèles des types d'homotopie":
http://www.math.univ-toulouse.fr/~dcisinsk/ast.pdf

Both are available in SMF's collection Astérisque.

I shall give you more details if nobody else shows up to explain the yoga (I myself have but a smattering of it).