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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:

  1. 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

  1. 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.)

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    $\begingroup$ This is an incarnation of the Whitehead theorem for model categories: a weak equivaience between cofibrant-fibrant objects is a homotopy equivalence. In topological spaces, CW complexes are cofibrant-fibrant. $\endgroup$ – Arun Debray May 5 '17 at 2:27
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The notions of weak equivalence are not 'the same' in your examples, rather they coincide on a certain class of objects, namely those that are 'cofibrant-fibrant' as mentioned in Arun's comment. Perhaps there is a more general question hidden in yours, however, as in both examples you have one notion of homotopy equivalence based on a cylinder or cocylinder functor and therefore feeling somewhat geometric, and one notion of 'quasi-isomorphism' relative to some functor, that I will write as $F$, to another context / category, $f:A\to B$ being a 'quasi-isomorphism' if $F(f)$ is an isomorphism. This is then a slightly more general question than yours in the context of model categories and is explored in Baues' work on algebraic homotopy.

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  • $\begingroup$ Ah I see, thanks. Do weak equivalences always coincide on the class of cofibrant-fibrant objects? $\endgroup$ – user316092 May 6 '17 at 11:55
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    $\begingroup$ Have a look at some of the books on Quillen's theory. On fibrant-cofibrant objects a weak equivalence has a homotopy inverse with homotopy given by a cylinder or cocylinder object (I paraphrase a bit here.) Your question is not quite well posed, as it is not clear what `coincide' refers to. $\endgroup$ – Tim Porter May 6 '17 at 12:36

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