The 2-Chu-Dialectica construction and the polycategory of multivariable adjunctions

Cheng, Gurski, and Riehl constructed a cyclic double multicategory of multivariable adjunctions.We show that the same information is carried by a poly double category, in which opposite categories are polycategorical duals.Moreover, this poly double category is a full substructure of a double Chu construction, whose objects are a sort of polarized category, and which is a natural home for 2-categorical dualities.We obtain the double Chu construction using a general "Chu-Dialectica" construction on polycategories, which includes both the Chu construction and the categorical Dialectica construction of de Paiva.The Chu and Dialectica constructions each impose additional hypotheses making the resulting polycategory representable (hence * -autonomous), but for different reasons; this leads to their apparent differences.Contents 1 Introductions 89 2 Presheaves of polycategories 104 3 Dimension (1, 1): the 2-Chu-Dialectica construction 108 4 Dimension (0, 1): the Dialectica construction 110 5 Dimension (0, 0): the Chu construction 113 6 Dimension (1, 0): the 2-Chu and double Chu constructions 116 7 Cyclic multicategories and parametrized mates 124 1. Introductions I have written two introductions to this paper, each of which can be read independently.If you are interested in Dialectica and Chu constructions, please continue with section 1.1;This material is based on research