Chu connections and back diagonals betweenQ-distributors

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.

Diagonals between $\mathcal{Q}$-distributors

“…In the case that C = Q is a (unital) quantale [24], the category D(Q) of diagonals of Q has been extensively studied in different frameworks; see, e.g., [12,6,38,13,15,23,35,36,29,14,37,10,17]. Indeed, D(Q) is a quantaloid [26]; that is, a category enriched in the symmetric monoidal closed category Sup of complete lattices and join-preserving maps.…”

“…where f ♮ and f ♮ are the graph and cograph of f , respectively, and ւ, ց are left and right implications in Q-Dist, respectively. The two equivalent characterizations of Chu transforms in (1.i) allow us to extend the category Q-Chu of Q-distributors and Chu transforms in two directions (see [29,Proposition 3.2.1]):…”

“…In the above diagram, ChuCon(Q-Dist) and B(Q-Dist), called the categories of Chu connections and back diagonals [29] of Q-Dist, dualize the constructions of Arr(Q-Dist) and D(Q-Dist), respectively. In fact, all the categories in (1.ii), expect Q-Chu, are actually quantaloids.…”

“…In fact, all the categories in (1.ii), expect Q-Chu, are actually quantaloids. It is already known that [29], and…”

“…(1) The canonical functor ChuCon(Q-Dist) op / / Q-Sup that leads to the equivalence B(Q-Dist) op ≃ Q-Sup (see [29,Proposition 3.3.1]) is constructed through fixed points of Isbell adjunctions [32], while the parallel functor Arr(Q-Dist) op / / Q-Sup (see [16,Proposition 5.1]) is constructed through fixed points of Kan adjunctions [32].…”