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. Models of Type Theory with Strict Equality. Phd thesis, School of Computer Science, University of Nottingham.). 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). A simple type system with two identity types. Unpublished note.), which constitutes one specific instance of a 2LTT.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 assumed by postulating that the inner and outer natural numbers types are 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 ((2015). Mathematical Structures in Computer Science 1–75.). Continuing this line of thought, we suggest a definition of $(\infty,1)$ -category and give some examples.

Transpension: The Right Adjoint to the Pi-type

“…• Given an auto-internal implementation, we can still pretend that we have a meta-internal situation by restricting ourselves to a subset of the language. But we automatically get a two-level type theory [Voe13,ACKS23], where we have access to non-fibrant types from within. (This does not prove conservativity of two-level type theory over the object system.)…”

Type-Theoretic Signatures for Algebraic Theories and Inductive Types

“…We describe models of 2LTT in the following. This is not the only possible way to present 2LTT; our approach differs from [ACKS19] in some ways. We will summarize the differences at the end of this section.…”

“…We digress a bit on a different way to present 2LTT. In the primary 2LTT reference [ACKS19], inner and outer layers are specified as follows. We have two different family structures on the base cwf, (Ty 0 , Tm 0 ) and (Ty 1 , Tm 1 ), and a morphism between them.…”

“…But so far it has not been solved, or proven impossible to solve. A key original motivation for 2LTT was precisely to allow construction of semisimplicial types [ACKS19].…”