First, an observation: Here are two examples of when different notions of weak equivalence in a category turn out to be the "same":

(Whitehead's Theorem) Two CW complexes are homotopy equivalent if and only if they are weakly homotopy equivalent (i.e. there is a map between them which induces an isomorphism on homotopy groups).

(An 'easy exercise') Two chain complexes of vector spaces are chain homotopy equivalent if and only if they are quasi-isomorphic (i.e. there is a map between them which induces an isomorphism on homology).

In both cases, one direction is immediate, the other direction takes more work. This prompts my questions:

**What are other examples of results that share this flavor? i.e. "Two objects are [weakly equivalent in way #1] if and only if they are [weakly equivalent in way #2]."**

and

**Is there a result in model category theory which says "Under such-and-such hypotheses, two notions of weak equivalences are the same" so that we can see the two examples above as corollaries?**

*(I'm completely new to model categories, so apologies for any/all naivety.)*