TITLE: Is there a high-concept explanation for why "simplicial" leads to "homotopy-theoretic"?
QUESTION [90 upvotes]: My (limited) understanding is that simplicial methods tend to be used whenever you want some kind of nontrivial homotopy theory -- for instance, to get a nice model structure, you use simplicial sets and not just plain sets; to make $\mathbb{A}^1$-homotopy work, you work with simplicial (pre?)sheaves and not just plain sheaves or schemes; to construct the cotangent complex (which if I understand correctly is a homotopical construction, hopefully a Quillen derived functor on the category of simplicial algebras), you use simplicial commutative rings.
But why does "simplicial" make everything work so well? For instance, a simplicial set is a contravariant functor $\Delta \to \mathbf{Sets}$ for $\Delta$ the simplex category: what is so wonderful about $\Delta$ that allows a model structure (and one, moreover, Quillen equivalent to topological spaces) appear?

REPLY [4 votes]: There are many ways in which the categories of simplicial sets and of  simplicial objects work very well, as mentioned above. More recently there has been a revival of the use of cubical sets, but with the additional  structure of connections, derived from  the monoid structures $\max,\min: I^2 \to I$. Dan Kan's first paper was cubical, but it was then realised that  cubical groups were not Kan complexes, and there was a serious problem with realisation of cartesian products. In 1996 A. Tonks proved cubical groups with connections are Kan and G. Maltsiniotis has proved that cubical sets with connections form a test category in the sense of Grothendieck. 
Thee is more on cubical sets in my answer to this mathoverflow question, in the book Nonabelian Algebraic Topology, and in this recent article. 
There are lots of areas which have been well worked over in the simplicial context but not in the cubical (with connection) context, e.g. cubical groups. So there is lots more evaluation work to be done.