Environments, Continuation Semantics and Indexed Categories

There have traditionally been two approaches to modelling environments, one by use of nite products in Cartesian closed categories , the other by use of the base categories of indexed categories with structure. Recently, there have been more general deenitions along both of these lines: the rst generalising from Cartesian to symmetric premonoidal categories, the second generalising from indexed categories with speciied structure to-categories. The added generality is not of the purely mathematical kind; in fact it is necessary to extend semantics from the logical calculi studied in, say, Type Theory to more realistic programming language fragments. In this paper, we establish an equivalence between these two recent notions. We then use that equivalence to study semantics for continuations. We give three category theoretic semantics for modelling continuations and show the relationships between them. The rst is given by a continuations monad. The second is based on a symmetric premonoidal category with a self-adjoint structure. The third is based on a-category with indexed self-adjoint structure. We extend our result about environments to show that the second and third semantics are essentially equivalent, and that they include the rst.