Construction of a crossed module of complexes from a precrossed module of complexes and a cat\(^1\)-complex from a precat\(^1\)-complex (Q2911204)

A crossed module \((C,G,\partial)\) of groups, defined by \textit{J. H. C. Whitehead} [Bull. Am. Math. Soc. 55, 453--496 (1949; Zbl 0040.38801)], is a group homomorphism \(\partial: C\rightarrow G\) with a (left) action of \(G\) on \(C\) satisfying the conditions CM1, \(\partial(^g c)=g\partial(c)g^{-1}\), and CM2, \( ^{\partial(c_1)} c_2=c_1c_2{c_1}^{-1}\), for all \(c,c_1,c_2\in C\) and \(g\in G\). A morphism of such crossed modules \((\mu,\eta): (C,G,\partial)\rightarrow (D,H,\delta)\) is a pair of group homomorphisms \(\mu: C\rightarrow D\) and \(\eta : G\rightarrow H\) such that \(\delta\mu=\eta\partial\) and \(\mu(^gc)= {^{\eta(g)}\mu(c)}\) for \(c\in C\) and \(g\in G\). So we get a category CModGrps of crossed modules. NEWLINESimilarly crossed modules of complexes are defined and the category of them is denoted by CModComp. A crossed module fulfilling condition CM1 only is called a precrossed module and similarly precrossed modules of complexes are defined. The category of them is denoted by PCModComp. NEWLINENEWLINENEWLINE On the other hand, a cat\(^1\)-group, defined by \textit{J.-L. Loday} [J. Pure Appl. Algebra 24, 179--202 (1982; Zbl 0491.55004)], is a group \(G\) along with two endomorphisms \(s,t: G\rightarrow G\) such that CAT1, \(ts=s\), \(st=t\), and CAT2, \([\operatorname{Ker}s, \operatorname{Ker}t]=1\). A morphism \(f: (G,s,t)\rightarrow (G',s',t')\) of cat\(^1\)-groups is a group homomorphism \(f: G\rightarrow G'\) such that \(s' f=f s\) and \(t' f=f t\). Hence we get a category Cat\(^1\)-Grps of cat\(^1\)-groups. Loday proved in [loc. cit.] that the categories CModGrps and Cat\(^1\)-Grps are equivalent. NEWLINEA cat\(^1\)-group fulfilling condition CAT1 only is called a precat\(^1\)-group. The notions of cat\(^1\)-group and precat\(^1\)-group are given in terms of complexes and the corresponding categories are denoted by Cat\(^1\)-Comp and PCat\(^1\)-Comp, respectively. NEWLINENEWLINENEWLINE In this paper a left-adjoint functor \(L: \mathrm{PCModComp}\rightarrow \mathrm{CModComp}\) to the forgetful functor \(F: \mathrm{CModComp}\rightarrow \mathrm{PCModComp}\) is obtained; and a similar result follows for the categories Cat\(^1\)-Comp and PCat\(^1\)-Comp.