The Delay Monad and Restriction Categories
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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.
KeywordsRestriction Categories Weak Bisimilarity Kleisli Category Monad Structure Kleisli Extension
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.
- 4.Capretta, V.: General recursion via coinductive types. Log. Methods Comput. Sci. 1(2), article 1 (2005)Google Scholar
- 5.Chapman, J., Uustalu, T., Veltri, N.: Quotienting the delay monad by weak bisimilarity. Math. Struct. Comput. Sci. (to appear)Google Scholar
- 6.Cockett, J.R.B., Guo, X.: Join restriction categories and the importance of being adhesive. Abstract of talk presented at CT 2007 (2007)Google Scholar
- 10.Danielsson, N.A.: Operational semantics using the partiality monad. In: Proceedings of 17th ACM SIGPLAN International Conference on Functional Programming, ICFP 2012, pp. 127–138. ACM, New York (2012)Google Scholar
- 11.Ésik, Z., Goncharov, S.: Some remarks on Conway and iteration theories. arXiv preprint arXiv:1603.00838 (2016)
- 13.Guo, X.: Products, joins, meets, and ranges in restriction categories. Ph.D. thesis, University of Calgary (2012)Google Scholar
- 18.Rosolini, G.: Continuity and effectiveness in topoi. DPhil. thesis, University of Oxford (1986)Google Scholar