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Is the distinction between quasi-isomorphism and `weak homotopy equivalence' ONLY that the first means inducing an isomorphism in homology and the second to an isomorphism of homotopy groups?

flag	asked Jul 4 2010 at 17:50 jim stasheff 1,386●2●10

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jim -- to rephrase your question: you give the definitions of two things and ask if the difference in ONLY that one definition defines one thing and the other definition defines the other thing. – algori Jul 4 2010 at 18:04

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Andrea- I do have to say that I think it's rather rude to accuse people of being Jim Stasheff impersonators without any evidence. I agree that it would be extraordinarily out of line to post as a well-known mathematician, but since I don't think we've had any serious cases of that, I'd give any poster use what appears to be a IRL name the benefit of the doubt. – Ben Webster♦ Jul 4 2010 at 19:12

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I seem to recall hearing or reading something like: If we restrict to, say, simply connected CW complexes, then X and Y are rationally weak homotopy equivalent (meaning there's a map inducing isomorphisms on homotopy groups tensor Q) if and only if C^*(X;Q) and C^*(Y;Q) are quasi-isomorphic dg algebras. Is this true? Maybe something like this is what Jim is asking about. – Kevin Lin Jul 4 2010 at 20:33

Kevin -- this is certainly true if one considers Sullivan's piecewise polynomial cochains. For singular cochains the "only if" part is obviously true, but the "if" part is subtle (presumably also true, but the details seem to have never appeared in literature). See mathoverflow.net/questions/9146/… – algori Jul 4 2010 at 20:48

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Guys -- let's remove this impersonator nonsense before more people see it. – algori Jul 6 2010 at 5:08

André Henriques · Answer 1 · 2010-10-12 08:50:38Z UTC

This is a terminology question.

I think that "weak homotopy equivalence" is mostly used for maps between topological spaces (as opposed to "weak equivalence", that is used in the much broader context of model categories).

The term "quasi-isomorphism", on the other hand, is typically used for (co)chain complexes, or (co)chain complexes equipped with extra structure.

So I would say that the main difference between those two terms is that they are used in different contexts.

quasi-isomorphism

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