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Homotopy types of finite étale spaces and generalized inflations

Anton Ayzenberg, Nadya Khoroshavkina

TL;DR

The paper develops a universal sheaf-theoretic framework for generalized inflations of simplicial posets along finite diagrams. By requiring the inflation sheaf to be inhabited and flabby, the authors extend the Björner–Björner–Welker poset fiber theorem to obtain a homotopy wedge decomposition of inflated spaces in terms of the base poset and the links of its simplices. This yields a unified proof of wedge decompositions for vertex-inflations and for cliques in multigraphs, and it recovers and broadens prior results (Wachs, AyzRukh) in a common language. The approach clarifies how étale spaces and finite-topology sheaves encode covering-space-type homotopy information, with potential implications for topological data analysis and quantum-contextuality models via a non-Abelian, combinatorial lens.

Abstract

Inflation of a simplicial complex $K$ is a construction well known in combinatorial topology. It replaces each vertex $i$ of $K$ with a finite number $n_i$ of its copies, and each simplex $\{i_0,\ldots,i_k\}$ with $n_{i_0}n_{i_1}\cdots n_{i_k}$ many copies so that the collection of vertex-copies is spanned by a simplex in the inflation if and only if their originals were spanned by a simplex in the original complex. The celebrated poset fiber theorem of Björner, Wachs, and Welker describes the homotopy type of such inflation in terms of homotopy types of $K$ and its links. In the current paper, we introduce more general inflations over simplicial posets: we replace each simplex with an arbitrary finite set of copies. The way how these sets patch together is specified by a commutative diagram, or, equivalently, a sheaf on the corresponding finite topology. The generalized inflation can be understood as étale space of such sheaf. We prove that, whenever this inflation sheaf is flabby, the poset fiber theorem still applies. We prove all results similar to those known for vertex inflations. We also cover the previous result of the first author about homotopy types of clique complexes of multigraphs.

Homotopy types of finite étale spaces and generalized inflations

TL;DR

The paper develops a universal sheaf-theoretic framework for generalized inflations of simplicial posets along finite diagrams. By requiring the inflation sheaf to be inhabited and flabby, the authors extend the Björner–Björner–Welker poset fiber theorem to obtain a homotopy wedge decomposition of inflated spaces in terms of the base poset and the links of its simplices. This yields a unified proof of wedge decompositions for vertex-inflations and for cliques in multigraphs, and it recovers and broadens prior results (Wachs, AyzRukh) in a common language. The approach clarifies how étale spaces and finite-topology sheaves encode covering-space-type homotopy information, with potential implications for topological data analysis and quantum-contextuality models via a non-Abelian, combinatorial lens.

Abstract

Inflation of a simplicial complex is a construction well known in combinatorial topology. It replaces each vertex of with a finite number of its copies, and each simplex with many copies so that the collection of vertex-copies is spanned by a simplex in the inflation if and only if their originals were spanned by a simplex in the original complex. The celebrated poset fiber theorem of Björner, Wachs, and Welker describes the homotopy type of such inflation in terms of homotopy types of and its links. In the current paper, we introduce more general inflations over simplicial posets: we replace each simplex with an arbitrary finite set of copies. The way how these sets patch together is specified by a commutative diagram, or, equivalently, a sheaf on the corresponding finite topology. The generalized inflation can be understood as étale space of such sheaf. We prove that, whenever this inflation sheaf is flabby, the poset fiber theorem still applies. We prove all results similar to those known for vertex inflations. We also cover the previous result of the first author about homotopy types of clique complexes of multigraphs.
Paper Structure (32 sections, 22 theorems, 34 equations, 1 figure)

This paper contains 32 sections, 22 theorems, 34 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Overview: sheaves and finite topologies
  3. Preorders and Notation
  4. Alexandrov Topologies
  5. Diagrams on partially ordered sets
  6. Sheaves on topologies
  7. Sheaves on Alexandrov topologies
  8. Trivial sheaf
  9. Induced sheaf, restriction of a sheaf
  10. Flabby sheaves
  11. Overview: combinatorial and homotopy topology
  12. Geometric realization of posets
  13. Homotopy colimits
  14. Completion along a diagram
  15. Étale spaces
  16. ...and 17 more sections

Key Result

Theorem 1.2

Assume that $f\colon N\to K$ is a surjective nondegenerate map of simplicial complexes (or simplicial posets). Assume moreover, that the corresponding sheaf $\mathcal{D}_f$ is flabby.

Figures (1)

  • Figure 1: From simplicial map to $\mathop{\mathrm{Sets}}\nolimits$-valued diagram over poset

Theorems & Definitions (69)

  • Theorem 1.2
  • Proposition 2.1
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Definition 2.9
  • Definition 2.10
  • Definition 2.11
  • ...and 59 more