Normalization graphs and Dold-Kan equivalences for generalized Reedy categories
Abstract
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.
Related
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BibTeX
@misc{li2026normalization,
title = {{Normalization graphs and Dold-Kan equivalences for generalized Reedy categories}},
author = {Liping Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38992},
eprint = {2609.38992},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38992},
}
RIS
TY - UNPB
TI - Normalization graphs and Dold-Kan equivalences for generalized Reedy categories
AU - Liping Li
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.38992
UR - https://arxiv.org/abs/2609.38992
AB - 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.
ER -