|
0
|
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? Thanks, |
|||||||||
|
|
2
|
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 http://pages.bangor.ac.uk/~mas010/nonab-a-t.html 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. |
|||
|
