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Cosheaf representations of relations and Dowker complexes

Cosheaf representations of relations and Dowker complexes The Dowker complex is an abstract simplicial complex that is constructed from a binary relation in a straightforward way. Although there are two ways to perform this construction—vertices for the complex are either the rows or the columns of the matrix representing the relation—the two constructions are homotopy equivalent. This article shows that the construction of a Dowker complex from a relation is a non-faithful covariant functor. Furthermore, we show that this functor can be made faithful by enriching the construction into a cosheaf on the Dowker complex. The cosheaf can be summarized by an integer weight function on the Dowker complex that is a complete isomorphism invariant for the relation. The cosheaf representation of a relation actually embodies both Dowker complexes, and we construct a duality functor that exchanges the two complexes. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Applied and Computational Topology Springer Journals

Cosheaf representations of relations and Dowker complexes

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Publisher
Springer Journals
Copyright
Copyright © The Author(s) 2021
ISSN
2367-1726
eISSN
2367-1734
DOI
10.1007/s41468-021-00078-y
Publisher site
See Article on Publisher Site

Abstract

The Dowker complex is an abstract simplicial complex that is constructed from a binary relation in a straightforward way. Although there are two ways to perform this construction—vertices for the complex are either the rows or the columns of the matrix representing the relation—the two constructions are homotopy equivalent. This article shows that the construction of a Dowker complex from a relation is a non-faithful covariant functor. Furthermore, we show that this functor can be made faithful by enriching the construction into a cosheaf on the Dowker complex. The cosheaf can be summarized by an integer weight function on the Dowker complex that is a complete isomorphism invariant for the relation. The cosheaf representation of a relation actually embodies both Dowker complexes, and we construct a duality functor that exchanges the two complexes.

Journal

Journal of Applied and Computational Topology – Springer Journals

Published: Mar 1, 2022

Keywords: Dowker complex; Cosheaf; Sheaf; Binary relation; 55U10

References

  • Whitney numbers of geometric lattices
    Bacławski, K
  • Topological methods
    Björner, A
  • Sparse Dowker nerves
    Brun, M; Blaser, N
  • A functorial Dowker theorem and persistent homology of asymmetric networks
    Chowdhury, S; Mémoli, F
  • Homology groups of relations
    Dowker, CH
  • The geometry of relations
    Minian, EG