(Co)Homology of crossed modules

Universal Central Extensions in Semi-Abelian Categories

“…The chain of reflections PXMod Ñ XMod Ñ AbXMod from precrossed modules to crossed modules to abelian crossed modules also corresponds to the situation considered in Section 3. Thus we regain results in [1,2,3,8].…”

(Co)homology of crossed modules with coefficients in a $\pi_1$-module

“…which relates the integral homology of the classifying space of a crossed module and the cotriple homology of [4].…”

The precise center of a crossed module

“…In [6], Carrasco, Cegarra and R.-Grandjeán proved that the category of crossed modules is an algebraic category, that is, that there is a tripleable forgetful functor from the category CM to the category of sets Set.…”

“…Recently Carrasco, Cegarra and R.-Grandjeán extended in [6] the EilenbergMacLane (co)homology of groups with their cotriple (co)homology of crossed modules. For each crossed module ðT; G; qÞ, abelian crossed module ðA; B; f Þ and for n d 1 they define the n-th homology crossed module H n ðT; G; qÞ of ðT; G; qÞ, and the n-th cohomology group H n ððT; G; qÞ; ðA; B; f ÞÞ of ðT; G; qÞ with coe‰cients in ðA; B; f Þ.…”

“…We begin in Section 2 by recalling known facts about the category of crossed modules and the definition of the homology, which can be found in detail in [6] or in [15]. We deduce some basic properties of projective crossed modules which will be used later.…”