A Note on n-Representations of Simplicial Groups
Abstract
For n≥1, we study representations of a simplicial group G in the full ∞-subcategory of simplicial vector spaces whose normalized homology is concentrated in degrees 0,…,n−1, retaining its full mapping spaces. The resulting functor category depends only on the n-truncated homotopy type of BG and admits projectively bifibrant strict diagram and simplicial group-algebra module models. If restriction along a specified map f:G→ H is an equivalence, then π0(f) is a group isomorphism; the proof uses ordinary extension and restriction of scalars. Yet for every n≥2 and prime q invertible in k, the nontrivial source K(ℤ/q,n−1) has the same n-representation category with vertex evaluation as the trivial group. The group-algebra augmentation is a weak equivalence, so the constant functor is also an equivalence for untruncated representations. Thus n-representations depend only on the n-truncation of BG, but need not determine that truncation, even together with vertex evaluation.
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BibTeX
@misc{pan2026note,
title = {{A Note on n-Representations of Simplicial Groups}},
author = {Zhangqi Pan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11576},
eprint = {2610.11576},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11576},
}
RIS
TY - UNPB
TI - A Note on n-Representations of Simplicial Groups
AU - Zhangqi Pan
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11576
UR - https://arxiv.org/abs/2610.11576
AB - For n≥1, we study representations of a simplicial group G in the full ∞-subcategory of simplicial vector spaces whose normalized homology is concentrated in degrees 0,…,n−1, retaining its full mapping spaces. The resulting functor category depends only on the n-truncated homotopy type of BG and admits projectively bifibrant strict diagram and simplicial group-algebra module models. If restriction along a specified map f:G→ H is an equivalence, then π_0(f) is a group isomorphism; the proof uses ordinary extension and restriction of scalars. Yet for every n≥2 and prime q invertible in k, the nontrivial source K(ℤ/q,n−1) has the same n-representation category with vertex evaluation as the trivial group. The group-algebra augmentation is a weak equivalence, so the constant functor is also an equivalence for untruncated representations. Thus n-representations depend only on the n-truncation of BG, but need not determine that truncation, even together with vertex evaluation.
ER -