Homomorphisms between standard modules of generalized Reedy categories

We develop a representation-theoretic approach to generalized Reedy categories through a systematic study of homomorphism spaces between standard modules. For a broad class of these categories, we provide a uniform, computable framework that reduces abstract homological constructions to elementary linear algebra and spectral graph theory via incidence matrices and morphism fibers. As a primary application, we establish a uniform extension of the Dold–Kan correspondence for categories arising from rooted trees, encompassing the categories of finite chains, finite sets and partial injections, and finite spiders. Crucially, this machinery unifies and provides a singular conceptual basis for several classic, seemingly disparate results across algebraic topology and representation theory, including Kuhn’s decomposition theorem for vector spaces and the Thévenaz–Webb semisimplicity theorem for Mackey functors.