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Chu connections and back diagonals are introduced as morphisms for distributors between categories enriched in a small quantaloid $mathcal{Q}$. 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 $mathcal{Q}$, the category of complete $mathcal{Q}$-categories and left adjoints is a retract of the dual of the category of $mathcal{Q}$-distributors and Chu connections, and it is dually equivalent to the category of $mathcal{Q}$-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.
Effectus theory is a relatively new approach to categorical logic that can be seen as an abstract form of generalized probabilistic theories (GPTs). While the scalars of a GPT are always the real unit interval [0,1], in an effectus they can form any
Cheng, Gurski, and Riehl constructed a cyclic double multicategory of multivariable adjunctions. We show that the same information is carried by a double polycategory, in which opposite categories are polycategorical duals. Moreover, this double poly
Doctrines are categorical structures very apt to study logics of different nature within a unified environment: the 2-category Dtn of doctrines. Modal interior operators are characterised as particular adjoints in the 2-category Dtn. We show that the
Monads can be interpreted as encoding formal expressions, or formal operations in the sense of universal algebra. We give a construction which formalizes the idea of evaluating an expression partially: for example, 2+3 can be obtained as a partial ev
We provide new categorical perspectives on Jacobs notion of hypernormalisation of sub-probability distributions. In particular, we show that a suitable general framework for notions of hypernormalisation is that of a symmetric monoidal category endow