发表机构
School of Mathematics and Statistics Huazhong University of Science and Technology（华中科技大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究单纯群的n-表示，证明其仅依赖BG的n-截断同伦型，给出相关模型与等价条件，发现非平凡群K(ℤ/q,n-1)与平凡群的n-表示范畴相同。
AI 中文摘要
对于n≥1，我们在单纯向量空间的全∞-子范畴中研究单纯群G的表示，该子范畴的归一化同调集中在次数0,…,n-1，且保留其全部映射空间。所得函子范畴仅依赖于BG的n-截断同伦型，且具有投射双余纤严图表及单纯群-代数模模型。若沿指定映射f:G→H的限制是等价，则π₀(f)是群同构；该证明使用标量的普通扩张与限制。然而对每个n≥2及在k中可逆的素数q，非平凡源K(ℤ/q,n-1)具有与平凡群相同的带顶点赋值的n-表示范畴。群-代数增广是弱等价，故常函子对未截断表示也是等价。因此n-表示仅依赖于BG的n-截断，但即便结合顶点赋值也未必决定该截断。
英文摘要
For $n\ge1$, we study representations of a simplicial group $G$ in the full $\infty$-subcategory of simplicial vector spaces whose normalized homology is concentrated in degrees $0,\ldots,n-1$, retaining its full mapping spaces. The resulting functor category depends only on the $n$-truncated homotopy type of $\mathbf{B}G$ and admits projectively bifibrant strict diagram and simplicial group-algebra module models. If restriction along a specified map $f:G\to H$ is an equivalence, then $π_0(f)$ is a group isomorphism; the proof uses ordinary extension and restriction of scalars. Yet for every $n\ge2$ and prime $q$ invertible in $k$, the nontrivial source $K(\mathbb Z/q,n-1)$ 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 $\mathbf{B}G$, but need not determine that truncation, even together with vertex evaluation.
Comments37 pages
