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We describe Galois connections which arise between two kinds of combinatorialstructures, both of which generalize trees with labelled leaves, and then applythose connections to a family of polytopes.  The graphs we study can be imbued with metric properties or associated tovectors. Famous examples are the Billera-Holmes-Vogtmann metric space ofphylogenetic trees, and the Balanced Minimal Evolution polytopes ofphylogenetic trees described by Eickmeyer, Huggins, Pachter and Yoshida.Recently the space of trees has been expanded to split networks by Devadoss andPetti, while the definition of phylogenetic polytopes has been generalized toencompass 1-nested phylogenetic networks, by Durell and Forcey. The firstGalois connection we describe is a reflection between the (unweighted) circularsplit networks and the 1-nested phylogenetic networks. Another Galoisconnection exists between certain metric versions of these structures.Reflection between the purely combinatorial posets becomes a coreflection inthe geometric case.  Our chief contributions here, beyond the discovery of the Galois connections,are: a translation between approaches using PC-trees and networks, a new way tolook at weightings on networks, and a fuller characterization of faces of thephylogenetic polytopes.

Galois connections for phylogenetic networks and their polytopes