Two-Level Type Theory and Applications

Abstract

We define and develop two-level type theory (2LTT), a version of Martin-Löf type theory which combines two different type theories. We refer to them as the inner and the outer type theory. In our case of interest, the inner theory is homotopy type theory (HoTT) which may include univalent universes and higher inductive types. The outer theory is a traditional form of type theory validating uniqueness of identity proofs (UIP). One point of view on it is as internalised meta-theory of the inner type theory. There are two motivations for 2LTT. Firstly, there are certain results about HoTT which are of meta-theoretic nature, such as the statement that semisimplicial types up to level $n$ can be constructed in HoTT for any externally fixed natural number $n$. Such results cannot be expressed in HoTT itself, but they can be formalised and proved in 2LTT, where $n$ will be a variable in the outer theory. This point of view is inspired by observations about conservativity of presheaf models. Secondly, 2LTT is a framework which is suitable for formulating additional axioms that one might want to add to HoTT. This idea is heavily inspired by Voevodsky's Homotopy Type System (HTS), which constitutes one specific instance of a 2LTT. HTS has an axiom ensuring that the type of natural numbers behaves like the external natural numbers, which allows the construction of a universe of semisimplicial types. In 2LTT, this axiom can be stated simply be asking the inner and outer natural numbers to be isomorphic. After defining 2LTT, we set up a collection of tools with the goal of making 2LTT a convenient language for future developments. As a first such application, we develop the theory of Reedy fibrant diagrams in the style of Shulman. Continuing this line of thought, we suggest a definition of (infinity,1)-category and give some examples.

58 pages; v5: typo fix and ref update

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Cited by in corpus (13)

• A General Framework for the Semantics of Type Theory
• All $(\infty ,1)$-toposes have strict univalent universes
• First-Order Logic with Isomorphism
• A type theory for synthetic $\infty$-categories
• A Rewriting Coherence Theorem with Applications in Homotopy Type Theory
• Transpension: The Right Adjoint to the Pi-type
• Synthetic fibered $(\infty ,1)$-category theory
• Coherence of strict equalities in dependent type theories
• Relative induction principles for type theories
• From Cubes to Twisted Cubes via Graph Morphisms in Type Theory
• The derivator of setoids
• Internal $\infty$-Categorical Models of Dependent Type Theory: Towards 2LTT Eating HoTT
• Types are Internal $\infty$-Groupoids