Diagonal Simplicial Tensor Modules and Algebraic $n$-Hypergroupoids

Florian Lengyel

Paper

Diagonal Simplicial Tensor Modules and Algebraic $n$-Hypergroupoids

Abstract

Let

be a commutative ring,

, and

with

. We attach to

a diagonal simplicial tensor module

whose

-simplices are functions on a cosimplicial index set

. This extends Quillen's diagonal on double simplicial groups to

commuting directions in

, compatibly with Dold--Kan normalization. We analyze the horn kernels

via ``missing indices'' and show that

if and only if

, independently of

. It follows that

is an algebraic

-hypergroupoid in the sense of Duskin--Glenn precisely when

, and that horn fillers in dimension

are non-unique precisely when

; in particular

is \emph{strict} in our sense precisely when

. A Horn Non-Degeneracy Lemma then shows that

and yields a decomposition

. A shift-and-truncate chain homotopy,

-equivariant and filtration-preserving, shows that

is contractible and forces the associated spectral sequence to collapse at

. When

is an infinite field

, we classify simplicial submodules generated by a single tensor via kernel sequences and a moduli map to a product of Grassmannians, obtaining an irreducible and unirational incidence variety.