A Model Category is, informally, a category where a "reasonable" notion of homotopy can be developed. I'm curious to know when two model categories are considered equivalent to each other. Thanks for any ideas, refs.
EDIT: My apologies ; what I meant to ask was on different ( inequivalent) model categories defined on the same category. Otherwise, like it was pointed out, the answer to my original question can be found in many books; even in Wikipedia.
The usual notion of equivalence for model categories is due to Quillen: given model categories $\mathcal{M}$ and $\mathcal{N}$, an adjunction $$F \dashv G : \mathcal{N} \to \mathcal{M}$$ is a Quillen equivalence if it is a Quillen adjunction (i.e. $F$ preserves cofibrations and trivial cofibrations, and $G$ preserves fibrations and trivial fibrations) such that the derived adjunction $$\mathbf{L} F \dashv \mathbf{R} G : \operatorname{Ho} \mathcal{N} \to \operatorname{Ho} \mathcal{M}$$ is an equivalence of categories. We say that two model categories are Quillen-equivalent if there is a zigzag of Quillen equivalences connecting them. This is all quite standard and can be found in any textbook on model categories.