Parameterizations and Fixed-Point Operators on Control Categories

The -calculus features both variables and names together with their binding mechanisms. This means that constructions on open terms are necessarily parameterized in two different ways for both variables and names. Semantically, such a construction must be modeled by a biparameterized family of operators. In this paper, we study these biparameterized operators on Selinger’s categorical models of the -calculus called control categories. The overall development is analogous to that of Lambek’s functional completeness of cartesian closed categories via polynomial categories. As a particular and important application of such consideration, we study the parameterizations of uniform fixed-point operators on control categories. We show a bijective correspondence between biparameterized fixed-point operators and nonparameterized ones under the uniformity conditions.