Normalization graphs and Dold-Kan equivalences for generalized Reedy categories

We introduce a combinatorial criterion for the projectivity of standard modules over generalized Reedy categories, formulated in terms of a finite bipartite graph whose normalized weightings determine normalization idempotents and hence projective splittings of standard modules. We apply this framework to recover several classical normalization constructions and Dold–Kan–type equivalences, and to obtain new Dold–Kan–type equivalences for categories of finite-dimensional affine, semilinear, and semiaffine spaces over a finite field. These equivalences also yield explicit descriptions of uniformly continuous representations of the corresponding infinite transformation monoids. Together with our previous work on Hom-spaces between standard modules, the results provide a combinatorial machinery, based mainly on graph theory and linear algebra, for establishing Dold–Kan–type equivalences for generalized Reedy categories.