A Relational Category of Birkhoff Polarities

TL;DR

This paper develops a categorical framework for polarities, based on Galois connections, establishing dualities, completeness, and morphism characterizations to deepen the mathematical understanding of these structures.

Contribution

It introduces a category of polarities with morphisms preserving Galois connections, extending Birkhoff's original idea and analyzing its structural properties.

Findings

Category of polarities is dual to complete meet semilattices.

The category is complete and has a factorization system.

It exhibits a star-autonomous structure.

Abstract

Garret Birkhoff observed that any binary relation between two sets determines a Galois connection between the powersets, or equivalently, closure operators on the powersets, or equivalently, complete lattices of subsets that are dually isomorphic. Referring to the duality of, say, points and lines in projective geometry, he named the binary relations as polarities. Researchers since then have used polarities (also known as formal contexts) as a convenient technical way to build complete lattices from ``found'' data. And so, various proposals for suitable morphisms between polarities have tended to have a particular application in mind. In this work, we develop the structure of a category of polarities and compatible relations, adopting Birkhoff's original simple idea that the structure of a polarity is its the Galois connection. Hence, morphisms must be relations that, in a reasonable sense, preserve Galois connections. In particular, the dual equivalence of the category to the category of complete meet semilattices, completeness of the category, characterization of epimorphisms and monomorphisms, an epi/mono factorization system, as well as the star-autonomous structure of the category, all arise by extending Birkhoff's original observation to morphisms.