The Delay Monad and Restriction Categories

Abstract

We continue the study of Capretta’s delay monad as a means of introducing non-termination from iteration into Martin-Löf type theory. In particular, we explain in what sense this monad provides a canonical solution. We discuss a class of monads that we call $\omega$-complete pointed classifying monads. These are monads whose Kleisli category is an $\omega$-complete pointed restriction category where pure maps are total. All such monads support non-termination from iteration: this is because restriction categories are a general framework for partiality; the presence of an $\omega$-join operation on homsets equips a restriction category with a uniform iteration operator. We show that the delay monad, when quotiented by weak bisimilarity, is the initial $\omega$-complete pointed classifying monad in our type-theoretic setting. This universal property singles it out from among other examples of such monads.