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]."
- 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.)