Why is Quillen equivalence used as the notion of equivalence of model categories?

So given two model categories $C,D$, $$L \dashv R:C \rightarrow D$$ is called a Quillen adjunction if $L$ preserves cofibrations and $R$ preserves fibrations (or some other equivalent formulations).

This induces a map in the homotopy categores: $$\Bbb L L \dashv \Bbb R R: Ho(C) \rightarrow Ho(D)$$

this is a Quillen equivalence if this adjunction is in fact an equivalence of categories.

What properties capture the idea that this is the right notion of "equivalence" by passing to homotopy category: rather than simply requiring adjoint equivalence of $C,D$?

References would help - I am aware of nlab post but am not satisfied.

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    $\begingroup$ What do you want to use model categories for, if not to study their homotopy categories? $\endgroup$ – Najib Idrissi Feb 10 at 11:32
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    $\begingroup$ If $C,D$ are equivalent as categories... they're equivalent as categories ! there would be no connection to the model structure or to homotopy theory, they'd just be essentially the same category. Your question sounds like : why do we say that two spaces are homotopy equivalent if they're isomorphic in the homotopy category; why do we go there and not just look at isomorphism in the category of spaces ? $\endgroup$ – Max Feb 10 at 11:40
  • $\begingroup$ in my opnion the big point of model theories is that they give quite good control over homotopy theory, as one can justuse fibrant cofibrant replacements. however, one might want to have even better models for the homotopy theory, respectively derived categories. Now it might be useful to find a different model structure that models it. Hence one looks for a more controlable category or model structure. For example in Louries book he constructs multiple such equivalences between the category of simplical sets with different model structures to control different properties infinity categories, $\endgroup$ – Enkidu Feb 10 at 15:52

In homotopy theory, we are interested to study objects up to weak equivalences. Recall that if $\mathcal{C}$ is a model category with a class of weak equivalence $\mathcal{W}$, then the associated homotopy category $\mathbf{Ho}(\mathcal{C})$ is the localization $\mathcal{C}[\mathcal{W}^{-1}]$ of $\mathcal{C}$ with respect to $\mathcal{W}$. So asking for an adjunction between two model categories $\mathcal{C}$ and $\mathcal{D}$ such that it induces an equivalence of categories in their localizations is a sensible thing to do.

For instance, Quillen showed that, up to weak equivalences, simplicial sets and topological spaces are "Quillen" equivalent (with suitable model categories). Topological spaces are far more complicated than simplicial sets and this results allows you to consider objects which are much more combinatorial, as long as you care about things up to homotopy.

However, this doesn't mean the definition is perfect. As you may have seen in the nlab post you linked, Dugger-Shipley have shown that equivalent homotopy categories doesn't necessarily imply there is a Quillen equivalence on the level of model categories.


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