The Delay Monad and Restriction Categories
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.
This research was supported by the Estonian Ministry of Education and Research institutional research grant IUT33-13 and the Estonian Research Council personal research grant PUT763.
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