Formalization of dependent type theory: The example of CaTT

by   Thibaut Benjamin, et al.

We present the type theory CaTT, originally introduced by Finster and Mimram to describe globular weak ω-categories, and we formalise this theory in the language of homotopy type theory. Most of the studies about this type theory assume that it is well-formed and satisfy the usual syntactic properties that dependent type theories enjoy, without being completely clear and thorough about what these properties are exactly. We use the formalisation that we provide to list and formally prove all of these meta-properties, thus filling a gap in the foundational aspect. We discuss the key aspects of the formalisation inherent to the theory CaTT, in particular that the absence of definitional equality greatly simplify the study, but also that specific side conditions are challenging to properly model. We present the formalisation in a way that not only handles the type theory CaTT but also all the related type theories that share the same structure, and in particular we show that this formalisation provides a proper ground to the study of the theory MCaTT which describes the globular, monoidal weak ω-categories. The article is accompanied by a development in the proof assistant Agda to actually check the formalisation that we present.



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1 Introduction

This article aims at presenting and formalising the foundations of a class of dependent type theories; the theory introduced by Finster and Mimram [catt] being the motivating example. This dependent type theory is designed to encode a flavor of higher categorical structures called weak -categories, and its semantics has been proved [benjamin2021globular] to be equivalent to a definition of weak -categories due to Maltsiniotis [maltsiniotis] based on an approach introduced by Grothendieck [pursuing-stacks]. However, none of the aforementioned articles about the theory provide a satisfying investigation of the type theoretic foundations, and simply assume that a lot of syntactic meta-properties are satisfied. This is not a shortcoming of those articles, but common practice in the type theory community since low-level descriptions are very lengthy and would make the articles essentially impossible to read. Thus it is usually accepted that such a description is possible and satisfies all the usually needed meta-theoretic properties can be proven by induction. Yet, there does not a general framework for presenting a dependent type theory, that would enforce the existence of low-level foundations together with the usual meta-theoretic properties. As a result, every new dependent type theory has to either be very specific about its foundations, or be vague enough and rely on the reader’s ability to infer the foundations and convince themselves that the meta-theoretic properties can be proven. This prevents a lot of results to be built from the ground up, and makes research about dependent type theory not readily available to experts.

The question of the choice of theoretical grounds to present a dependent type theory is also an interesting question in its own rights. It has been a long-standing goal to define type theory within type theory [chapman2009type], and significant progress has been met in this direction, but there are still open problems when it comes to fully embracing proof-relevance. In particular, it has been theorised that homotopy type theory (HoTT) [hottbook] should be completely definable within itself, but providing such a definition is still one of the main open problems of HoTT. Notably, this definition raises an important coherence problem that is reminiscent in its nature of the problems encountered in defining higher structures. Altenkirch and Kaposi [altenkirch2016type] have made progress towards solving this issue by using quotient inductive inductive types, a type theoretic construct whose semantics is not completely understood.

In this article, we present a proof of concept for using the language of HoTT as a meta-theory for defining and studying dependent type theories. Our main objective is the theory , so our formalisation within HoTT also provide a much-needed foundation to this theory. Moreover this theory is particularly simple, since it does not have definitional equality, making it an ideal candidate to focus on one challenge posed by the use of HoTT. We first give a quick informal presentation of the theory , in order to assert our goal. This presentation relies on a simpler type theory called which describes globular sets. We then discuss the formalisation of this theory, along with its meta-theoretic properties. Next, we move on to the formalisation of dependent type theories whose shapes are described by the theory , that we call globular type theories, and whose is our primary example. Finally we present the theory and show how it can be formalised in the framework of globular type theories. We prove some of the interesting meta-properties of this theory with a particular emphasis on the ones that are important to understand its semantics and that are used in other articles without full proofs [catt, benjamin2021globular]. All our definitions and proofs are accompanied by a formalisation in the theorem prover 111, (to be consistent with the language of HoTT, we also deactivated the use of the axiom K in ).

2 Introduction to the theory

We present here the type theory in order to motivate our formalisation work. The introduction we provide here focuses mostly on the syntactic aspects of the theory, but we also provide some intuition to the semantics, to help situate the theory in a broader picture. We refer the reader to existing articles [catt, benjamin2021globular] for more in-depth discussions about the semantics of this type theory. The first presentation we provide here is informal, and serves as a guideline for the foundations that we are developing.

2.1 General setup for dependent type theories

All along this article, we only consider dependent type theories which support the contraction, exchange and weakening rules, so we simply refer to them as type theories and assume those rules implicitly. Those theories are centred around four kinds of object, that we introduce here along with corresponding notations

The type theories we are interested in do not have definitional equalities, so these are the only judgements we do consider here.

2.2 The theory

We first introduce the theory which is simpler than the theory and serves as a basis on which this theory relies. In the theory there are no term constructors, hence the only terms are the variables. There are two type constructors, that we denote and . Those two constructors are subject to the following introduction rules