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Could you tell me how are two crossed modules quasi-isomorphic. And I have known a result: Let $\mu: M \rightarrow G$ and $\mu': M' \rightarrow G'$ are isomorphic, then the integral homology of them are the same. How to prove this result?

Could you show me related materials?


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Please see the FAQ, in particular what sort of questions are appropriate for MO. – David Roberts Apr 2 '12 at 0:40
In particular, I believe this question would be better placed at (and please read their FAQ before posting there) – David Roberts Apr 2 '12 at 2:50
I'm sorry. It was the first time I posted my question. I will see FAQ. Thanks. – Sapus Apr 4 '12 at 16:30

Even if David Roberts is right, I'll give an answer; more details my be found in the book

R. Brown, P.J. Higgins, R. Sivera, Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids, EMS Tracts in Mathematics Vol. 15, 703 pages. (August 2011).

of which a pdf may be downloaded from

Any crossed module $\mu: M \to P$ has a classifying space $X=B(\mu)$ whose homotopy groups are 0 above dimension 2 and in dimensions 1 and 2 are respectively Coker $\mu$ and Ker $\mu$. A morphism $f$ of crossed modules is a weak equivalence iff the induced map of classifying spaces is a weak equivalence. The homology of a crossed module is defined as the homology of its classifying space. The result asked for is easily obtained by passing to universal covers.

For more information see also

{Ellis, Graham J.}, TITLE = {Homology of {$2$}-types}, JOURNAL = {J. London Math. Soc. (2)}, VOLUME = {46}, YEAR = {1992}, NUMBER = {1}, PAGES = {1--27},

and the subsequent correction.

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Thank you very much. – Sapus Apr 4 '12 at 16:29

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