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.
Keywords: multivariable adjunction, polycategory, Dialectica construction, Chu construction
2010 MSC: 18D10,18D05
Theory and Applications of Categories, Vol. 35, 2020, No. 4, pp 89-136.