Models of Type Theory with Strict Equality

Abstract

This thesis introduces the idea of two-level type theory, an extension of Martin-Löf type theory that adds a notion of strict equality as an internal primitive. A type theory with a strict equality alongside the more conventional form of equality, the latter being of fundamental importance for the recent innovation of homotopy type theory (HoTT), was first proposed by Voevodsky, and is usually referred to as HTS. Here, we generalise and expand this idea, by developing a semantic framework that gives a systematic account of type formers for two-level systems, and proving a conservativity result relating back to a conventional type theory like HoTT. Finally, we show how a two-level theory can be used to provide partial solutions to open problems in HoTT. In particular, we use it to construct semi-simplicial types, and lay out the foundations of an internal theory of $(\infty ,1)$-categories.

References in corpus (4)

• Higher Topos Theory
• The identity type weak factorisation system
• On the homotopy groups of spheres in homotopy type theory
• Higher Homotopies in a Hierarchy of Univalent Universes

Cited by in corpus (6)

• Two-Level Type Theory and Applications
• A Category Theoretic View of Contextual Types: from Simple Types to Dependent Types
• Coherence of strict equalities in dependent type theories
• A Role for Dependent Types in Haskell (Extended version)
• From Cubes to Twisted Cubes via Graph Morphisms in Type Theory
• Internal $\infty$-Categorical Models of Dependent Type Theory: Towards 2LTT Eating HoTT