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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2512.10281 |
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| _version_ | 1866918276218486784 |
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| author | Lengyel, Florian |
| author_facet | Lengyel, Florian |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_10281 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Diagonal Simplicial Tensor Modules and Algebraic $n$-Hypergroupoids Lengyel, Florian Combinatorics 18G30 (Primary) 5U10, 14M15, 05E45, 15A69 (Secondary) 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. |
| title | Diagonal Simplicial Tensor Modules and Algebraic $n$-Hypergroupoids |
| topic | Combinatorics 18G30 (Primary) 5U10, 14M15, 05E45, 15A69 (Secondary) |
| url | https://arxiv.org/abs/2512.10281 |