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Cosystolic Expansion of Sheaves on Posets with Applications to Good 2-Query Locally Testable Codes and Lifted Codes

Uriya A. First, Tali Kaufman

TL;DR

This work develops a local-to-global framework for cosystolic expansion of sheaves on finite posets, extending known local criteria from simplicial complexes to general posets and arbitrary sheaves. A central technical result (the main cosystolic-expansion criterion) shows that local coboundary expansion together with skeleton-expansion data from no-intersection graphs yields global cosystolic expansion and cocycle-distance guarantees; the authors further provide a refined zero-th case with explicit constants. Two primary applications follow: (i) constructing good $2$-query locally testable codes by viewing them as cocycle codes over square complexes, linking their testability to cosystolic expansion; (ii) giving a purely local criterion for the local testability of lifted codes via a $2$-layer lifted-code structure, avoiding global assumptions. These results unify and extend prior advances in cosystolic expansion, testability of codes, and high-dimensional expanders, with practical implications for constructing efficient LTCs and understanding local constraints in lifted-code frameworks. The framework leverages poset-based no-intersection graphs, intersection profiles, and refined notions of skeleton-expansion to enable robust local-to-global conclusions across non-simplicial settings.

Abstract

We study sheaves on posets, showing that cosystolic expansion of such sheaves can be derived from local expansion conditions of the sheaf and the poset (typically a high dimensional expander). When the poset at hand is a cell complex, a sheaf on it may be thought of as generalizing coefficient groups used for defining homology and cohomology, by letting the coefficient group vary along the cell complex. Previous works established local criteria for cosystolic expansion only for simplicial complexes and with respect to constant coefficients. Cosystolic expansion of sheaves is related to property testing. We use this relation and our local criterion for cosystolic expansion to give two applications to locally testable codes (LTCs). First, we show the existence of good $2$-query LTCs. These codes are related to the recent good $q$-query LTCs of Dinur et. al and Panteleev-Kalachev, being the formers' so-called line codes, but we get them from a new, more illuminating perspective, namely, by realizing them as cocycle codes of sheaves over posets. We then derive their good properties directly from our criterion for cosystolic expansion. Second, we give a local criterion for a a lifted code (with some auxiliary structure) to be locally testable. This improves on a previous work of Dikstein et. al, where it was shown that one can obtain local testability of lifted codes from a mixture of local and global conditions.

Cosystolic Expansion of Sheaves on Posets with Applications to Good 2-Query Locally Testable Codes and Lifted Codes

TL;DR

This work develops a local-to-global framework for cosystolic expansion of sheaves on finite posets, extending known local criteria from simplicial complexes to general posets and arbitrary sheaves. A central technical result (the main cosystolic-expansion criterion) shows that local coboundary expansion together with skeleton-expansion data from no-intersection graphs yields global cosystolic expansion and cocycle-distance guarantees; the authors further provide a refined zero-th case with explicit constants. Two primary applications follow: (i) constructing good -query locally testable codes by viewing them as cocycle codes over square complexes, linking their testability to cosystolic expansion; (ii) giving a purely local criterion for the local testability of lifted codes via a -layer lifted-code structure, avoiding global assumptions. These results unify and extend prior advances in cosystolic expansion, testability of codes, and high-dimensional expanders, with practical implications for constructing efficient LTCs and understanding local constraints in lifted-code frameworks. The framework leverages poset-based no-intersection graphs, intersection profiles, and refined notions of skeleton-expansion to enable robust local-to-global conclusions across non-simplicial settings.

Abstract

We study sheaves on posets, showing that cosystolic expansion of such sheaves can be derived from local expansion conditions of the sheaf and the poset (typically a high dimensional expander). When the poset at hand is a cell complex, a sheaf on it may be thought of as generalizing coefficient groups used for defining homology and cohomology, by letting the coefficient group vary along the cell complex. Previous works established local criteria for cosystolic expansion only for simplicial complexes and with respect to constant coefficients. Cosystolic expansion of sheaves is related to property testing. We use this relation and our local criterion for cosystolic expansion to give two applications to locally testable codes (LTCs). First, we show the existence of good -query LTCs. These codes are related to the recent good -query LTCs of Dinur et. al and Panteleev-Kalachev, being the formers' so-called line codes, but we get them from a new, more illuminating perspective, namely, by realizing them as cocycle codes of sheaves over posets. We then derive their good properties directly from our criterion for cosystolic expansion. Second, we give a local criterion for a a lifted code (with some auxiliary structure) to be locally testable. This improves on a previous work of Dikstein et. al, where it was shown that one can obtain local testability of lifted codes from a mixture of local and global conditions.
Paper Structure (73 sections, 72 theorems, 256 equations, 1 table)

This paper contains 73 sections, 72 theorems, 256 equations, 1 table.

Table of Contents

  1. Overview
  2. General
  3. Posets
  4. Sheaves
  5. Brief History and Related Notions.
  6. Sheaf Cohomology.
  7. Locally Testable Codes From Sheaves on Posets
  8. Coboundary Expansion
  9. A Criterion For Cosystolic Expansion of Sheaves
  10. First Application: Good $2$-Query LTCs
  11. Second Application: A Local Criterion for Local Testablity of $2$-Layer Lifted Codes
  12. $2$-Layer Lifted Codes.
  13. Local Testability of $2$-Layer Lifted Codes.
  14. Conclusion
  15. Structure of This Paper
  16. ...and 58 more sections

Key Result

Theorem 1.2

For every $k\in\mathbb{N}$, $F\in\mathbb{N}$, $L\in[1,\infty)$ and $B\in\mathbb{R}_+$, there are constants $K,K'\in (0,1]$ such that the following holds: Let $X$ be a $d$-poset ($d\geq k+2$) with $L(X)\leq L$ and such that every $(k+2)$-face of $X$ has at most $F$-subfaces, let ${\mathcal{F}}$ be a Then $\mathop{\mathrm{cse}}\nolimits_k({\mathcal{F}})\geq K'(K\varepsilon)^{2^{k+2}-1}$ and $\matho

Theorems & Definitions (217)

  • Remark 1.1
  • Theorem 1.2: Simplified; see Theorem \ref{['TH:lgp-simple-version']}
  • Theorem 1.3: Criterion for $0$-Cosystolic Expansion; simplified; see Theorem \ref{['TH:lgp-zero-detailed']}
  • Theorem 1.4: Simplified; see Theorem \ref{['TH:ltc-square']}
  • Remark 1.5
  • Theorem 1.6: Simplified; see Theorem \ref{['TH:two-layer-local']}
  • Example 2.1: Cayley Graph
  • Example 2.2
  • Proposition 2.3: First_2022_cbe_sheaves_on_graphs
  • Remark 2.4: Codes with Varying Alphabets
  • ...and 207 more