A Note on n-Representations of Simplicial Groups

Abstract

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 $\pi_0(f)$ 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.