Standard modules and Dold-Kan correspondences for generalized Reedy categories

Abstract

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 Th\'evenaz--Webb semisimplicity theorem for Mackey functors.