We develop domain theory in constructive univalent foundations without Voevodsky’s resizing axioms. In previous work in this direction, we constructed the Scott model of PCF and proved its computational adequacy, based on directed complete posets (dcpos). Here we further consider algebraic and continuous dcpos, and construct Scott’s D∞ model of the untyped λ-calculus. A common approach to deal with size issues in a predicative foundation is to work with information systems or abstract bases or formal topologies rather than dcpos, and approximable relations rather than Scott continuous functions. Here we instead accept that dcpos may be large and work with type universes to account for this. For instance, in the Scott model of PCF, the dcpos have carriers in the second universe U1 and suprema of directed families with indexing type in the first universe U0. Seeing a poset as a category in the usual way, we can say that these dcpos are large, but locally small, and have small filtered colimits. In the case of algebraic dcpos, in order to deal with size issues, we proceed mimicking the definition of accessible category. With such a definition, our construction of Scott’s D∞ again gives a large, locally small, algebraic dcpo with small directed suprema.
|Title of host publication||29th EACSL Annual Conference on Computer Science Logic, CSL 2021|
|Editors||Christel Baier, Jean Goubault-Larrecq|
|Number of pages||18|
|Publication status||Published - 13 Jan 2021|
|Event||29th EACSL Annual Conference on Computer Science Logic - Held Online Due to COVID-19, Ljubljana, Slovenia|
Duration: 25 Jan 2021 → 28 Jan 2021
|Name||Leibniz International Proceedings in Informatics, LIPIcs|
|Conference||29th EACSL Annual Conference on Computer Science Logic|
|Abbreviated title||CSL 2021|
|Period||25/01/21 → 28/01/21|
Bibliographical notePublisher Copyright:
© Tom de Jong and Martín Hötzel Escardó.
- Domain theory
- Univalent foundations
ASJC Scopus subject areas