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 Thévenaz--Webb semisimplicity theorem for Mackey functors.
Related
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BibTeX
@misc{li2026standard,
title = {{Standard modules and Dold-Kan correspondences for generalized Reedy categories}},
author = {Liping Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.12499},
eprint = {2607.12499},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.12499},
}
RIS
TY - UNPB
TI - Standard modules and Dold-Kan correspondences for generalized Reedy categories
AU - Liping Li
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2607.12499
UR - https://arxiv.org/abs/2607.12499
AB - 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.
ER -