Diagonal Simplicial Tensor Modules and Algebraic $n$-Hypergroupoids

TL;DR

This paper introduces diagonal simplicial tensor modules associated with hypergroupoids, analyzing their horn kernels, algebraic properties, and spectral sequences, with applications to tensor moduli over infinite fields.

Contribution

It extends Quillen's diagonal construction to k-fold simplicial modules, characterizes hypergroupoid conditions, and provides explicit homotopies and moduli space descriptions.

Findings

Horn kernels vanish unless k ≥ p

Modules are algebraic n-hypergroupoids iff k ≤ n

Spectral sequence collapses at E_1

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.