I could not come up with a better title for my question.
What I am asking is this (preemptive excuses to all experts in homotopy theory for naivetes of all kinds you may find herein):
the category of simplicial sets has always been to me something distinguished, on multiple counts:
to begin with, it is simple to describe, almost childish, yet it has a seemingly unfathomable richness.
Its homotopy theory, although usually presented via the geometric realization functor (which is historically correct), is in fact entirely self-contained, and purely combinatorial in character.
Also, simplicial sets plays a very special role in category theory (after all, categories are just some almost trivial example of simplicial sets) and, even more important, in higher dimensional cats.
Somehow, I have the lingering feeling that, in abstract homotopy theory, simplicial sets (or, more properly simplicial objects in some ambient category ) should be, at least for some suitable notion of "regularity" of homotopy theories, paradigmatic. I am thinking of something like:
If an homotopy theory is "regular...." (whatever that may mean, fill the dots, the "regular" would stand for combinatorial in essence), then it is representable in the homotopy theory of simplicial objects for some ambient category.
Am I dreaming or there is something along these lines?