No abstract

Chu connections and back diagonals are introduced as morphisms for distributors between categories enriched in a small quantaloid Q. These notions, meaningful for closed bicategories, dualize the constructions of arrow categories and the Freyd completion of categories. It is shown that, for a small quantaloid Q, the category of complete Q-categories and left adjoints is a retract of the dual of the category of Q-distributors and Chu connections, and it is dually equivalent to the category of Q-distributors and back diagonals. As an application of Chu connections, a postulation of the intuitive idea of reduction of formal contexts in the theory of formal concept analysis is presented, and a characterization of reducts of formal contexts is obtained.

Inspired by the theory of apartness relations of Scott, we establish a positive theory of dissimilarity valued in an involutive quantale Q without the aid of negation. It is demonstrated that a set equipped with a Q-valued dissimilarity is precisely a symmetric category enriched in a subquantaloid of the quantaloid of back diagonals of Q. Interactions between Q-valued dissimilarities and Q-valued similarities (which are equivalent to Q-valued equalities in the sense of Höhle-Kubiak) are investigated with the help of lax functors. In particular, it is shown that similarities and dissimilarities are interdefinable if Q is a Girard quantale with a hermitian and cyclic dualizing element.