Diagonal Simplicial Tensor Modules and Algebraic $n$-Hypergroupoids

Abstract

Let $A$ be a commutative ring, let $k\in\mathbb{Z}^+$, and let $\vec{s}=(n_1,\dots,n_k)\in(\mathbb{Z}^+)^k$ with $n=\min_a(n_a)-1$. We attach to $\vec{s}$ a diagonal simplicial tensor module $X_\bullet(\vec{s};A)$ whose $p$-simplices are functions on a cosimplicial index set $I_p(\vec{s})\subseteq \mathbb{N}^k$. This extends Quillen's diagonal on double semi-simplicial groups: $X_\bullet(\vec{s};A)$ is obtained by restricting a $k$-fold simplicial $A$-module along the diagonal $p\mapsto(p,\ldots,p)$. Using a ``missing indices'' description of face kernels, we compute the horn kernels $R_{p,j}(X)$ and show that $R_{p,j}(X)\neq 0$ if and only if $k\ge p$, independently of $j$. Consequently, $X_\bullet(\vec{s};A)$ is an algebraic $n$-hypergroupoid in the sense of Duskin (1979) and Glenn (1982) if and only if $k\le n$, and horn fillers in dimension $n$ are non-unique if and only if $k\ge n$; in particular it is strict precisely when $k=n$. A Horn Non-Degeneracy Lemma shows that, for $p\ge 1$, $R_{p,j}(X)\cap D_p(X)=\{0\}$ and yields a decomposition $X_p=R_{p,j}(X)\oplus D_p(X)$. An explicit shift-and-truncate chain homotopy, equivariant under $\operatorname{Stab}(\vec{s})$ and compatible with a natural filtration, contracts $X_\bullet(\vec{s};A)$ and forces the associated spectral sequence to collapse at $E_1$. When $A$ is an infinite field $K$, we study simplicial submodules generated by a single tensor via kernel sequences and a moduli map to a product of Grassmannians. The moduli map image is an irreducible and unirational constructible subset of a determinantal incidence variety.