The Wikipedia article about Acyclic models notices that the way that they were used in the proof of the Eilenberg–Zilber theorem laid the foundation stone to the idea of the model category.
Could somebody explain what exactly in the concept of acyclic models gave rise to considering/developing the concept of model category? What is precisely the core relation between the both concepts?
Recall that the intuition behind model categories is to "extract the essence" (e.g., containing certain distinguished classes of maps), which a category should intrinsically have in order to be "amenable" to constructing a homotopy theory from it.
The acyclic model is a subcategory $\mathcal{M} \subset \mathcal{K}$ which allows (if certain technical assumptions are satisfied) to develop a criterion when for two covariant functors $F,G: \mathcal{K} \to \operatorname{Comp}(R)$ (the latter is the category of complexes of $R$-modules for some ring $R$) and two natural transformations $f,g: F \to G$ there exists a chain homotopy between $f$ and $g$ presuming one "knows" what happens "in $0$-th layer", i.e., the zeroth homology of associated complexes which are "sufficiently well controlled" by restriction to $\mathcal{M}$. A kind of "lifting structure", which "knows" enough at $0$-layer about these functors $F$, $G$, to know about them at "all layers".
At least that's the rough idea how I understood the philosophy of acyclic models.
Now back to the question I posed at the beginning: In which way do acyclic models concretely motivate the idea behind a model category?
A naïve guess: Do the acyclic models $\mathcal{M} \subset \mathcal{K}$ in some sense or after appropriate modification provide archetypical examples of what later becomes axiomatically characterized as model category? If yes, can a model category be understood as attempt to axiomatize the "structural essence" of what structure an acyclic model should at least have, in order to have enough structure to be useful as an acyclic model for certain unspecified category $\mathcal{K}$ containing it?
Formulated in other words, could one say that $\mathcal{M}$ is a model category if and only if there exist a category $\mathcal{K}$ containing it, functors $F$, $G$ as above, such that $\mathcal{M}$ works as an acyclic model for it? (…just a naïve guess)?
Or in which other way does the idea of model categories arise from the concept of acyclic models?
