DRAFT: work in progress, not for citation, comments solicited Attributes as dual types: a unification of presheaves and Chu spaces

We propose a framework for types and attributes that, instead of reducing attribute values such as red to types or making attributes columns of relations, organizes attributes as dual types and attribute values as morphisms from types to attributes. We formalize this organization syntactically via bimodules, and semantically via communes over bimodules. Communes generalize presheaves (for types) by pairing them with dual presheaves (for attributes), and generalize Chu spaces by stratifying their points and states with types and attributes respectively. Bimodules and commune categories arise as respectively initial and final bigenerated categories, analogously to the Kleisli/Eilenberg-Moore relationship. The commune categories, including Set, provide naturally arising instances of Cockett and Seely’s notion of linearly distributive category without units.