This is a follow-up of this question, where the definition of a quasi-abelian crossed module was given. Namely, a crossed module $\partial\colon F\to G$ is quasi-abelian if the embedding $\partial_Z\hookrightarrow \partial$ of its center into it is a quasi-isomorphism.

Question 1. Assume that $\partial\colon F\to G$ is a crossed module which can be related by a chain of quasi-isomorphisms to a quasi-abelian crossed module. Then $\pi_1(\partial):={\rm coker}\ \partial$ is abelian and acts trivially on $\pi_2(\partial):=\ker\partial$ (is the assertion about the action true?). What else can be said about $\partial$?

Question 2. Assume that $\partial$ and $\partial'$ are two quasi-abelian crossed modules which can be related by a chain of quasi-isomorphisms. Is it true that their centers $\partial_Z$ and $\partial'_Z$ are isomorphic in the derived category of complexes of abelian groups, i.e. can be related by a chain of quasi-isomorphisms of complexes of abelian groups?

Both questions are motivated by cohomology with coefficients in crossed modules.


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