Access the full text.
Sign up today, get DeepDyve free for 14 days.
Loading next page...
/lp/springer-journals/cosheaf-representations-of-relations-and-dowker-complexes-GNepRIRHHV
References (23)
-
C. Dowker (1952)
HOMOLOGY GROUPS OF RELATIONSAnnals of Mathematics, 56
-
COSHEAF REPRESENTATIONS OF RELATIONS AND DOWKER COMPLEXES 33 -
Lars Salbu (2019)
Dowker’s Theorem by Simplicial Sets and a Category of 0-Interleavings -
K. Baclawski (1975)
Whitney Numbers of Geometric LatticesAdvances in Mathematics, 16
-
J. Adámek, H. Herrlich, G. Strecker (1990)
Abstract and Concrete Categories - The Joy of Cats -
There is a duality functor that exchanges the cosheaf’s base space and space of global cosections (Theorem 7) -
D. Rydeheard, R. Burstall (1988)
Computational category theory -
R. Živaljević (2004)
Topological Methods -
Žiga Virk (2019)
Rips Complexes as Nerves and a Functorial Dowker-Nerve DiagramMediterranean Journal of Mathematics, 18
-
R. Ghrist (2014)
Elementary Applied Topology -
Elias Minian (2007)
The Geometry of RelationsOrder, 27
-
(1975)
Bac(cid:32)lawski -
Michael Robinson (2014)
Topological Signal Processing -
The space of global cosections of the cosheaf is the dual Dowker complex for the relation (Theorem 6) -
The existence of faithful functors that render the relation into a cosheaf (Theorem 4 and Corollary 4) or sheaf (Theorem 5), whose (co)stalks determine the total weight function, -
M. Brun, N. Blaser (2018)
Sparse Dowker nervesJournal of Applied and Computational Topology
-
(2021)
Sheaf theoryMathematical Surveys and Monographs
-
Kristopher Ambrose, Steve Huntsman, Michael Robinson, Matvey Yutin (2020)
Topological Differential TestingArXiv, abs/2003.00976
-
(2014)
Functors on posets left Kan extend to cosheaves, arXiv:1907.09416 -
Samir Chowdhury, F. Mémoli (2016)
A functorial Dowker theorem and persistent homology of asymmetric networksJournal of Applied and Computational Topology, 2
-
Samir Chowdhury, F. Mémoli (2016)
Persistent Homology of Asymmetric Networks: An Approach based on Dowker Filtrations -
J. Curry (2013)
Sheaves, Cosheaves and ApplicationsarXiv: Algebraic Topology
-
Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations
- 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