Homomorphisms between standard modules of generalized Reedy categories
arXiv:2607.12499
The paper develops a representation‑theoretic framework for generalized Reedy categories by studying homomorphisms between their standard modules, reducing many constructions to linear algebra and spectral graph theory, and applying this to a unified extension of the Dold–Kan correspondence and related results in algebraic topology and Mackey functor theory.
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 Thévenaz--Webb semisimplicity theorem for Mackey functors.