  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.
