On homology with coefficients and generalized inductions

In group representations several inductions given by tensoring with appropriate bimodules may be reconstructed via homology of $G$-posets with $G$-equivariant coefficients. For this purpose, we need various local categories of a finite group $G$, which afford the coefficients. Consequently, the functors among local categories give rise to the homology constructions naturally, and may be used to reformulate some existing results, as well as to deduce new statements.