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Suppose we are given a crossed-homomorphism $\phi:G\to A$ (and an action $\alpha$ of $G$ on $A$)

$\phi(ab)=\phi(a)+\alpha(a)(\phi(b))$. Now, unless the action is trivial, this is not a homomorphism and therefore it is not a morphism in the category of groups.

Is there a way to interpret crossed-homomorphisms as morphisms in some category?

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  • $\begingroup$ Here's a hint: is the composite of two crossed homomorphisms a crossed homomorphism? $\endgroup$
    – David Roberts ♦
    Commented May 9 at 6:57
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    $\begingroup$ @DavidRoberts Here's a hint: it actually is, if you manage to write a correct definition (in spare time between giving hints). The right object would be the category of functors between two crossed modules (i. e. 2-groupoids). Classical crossed homomorphisms correspond to maps out of identity functor, or, equivalently, sections of $A \rtimes G \to G$ which are group homomorphisms. $\endgroup$
    – Denis T
    Commented May 9 at 7:28
  • $\begingroup$ Hi @DenisT, thank you very much for your comment! Could you please expand on your comment with some more details in an answer? For example, it's not clear to me what you mean by sections $\endgroup$
    – rick
    Commented May 9 at 13:28

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