[Paper Review] Classifying Types

Egbert Rijke|arXiv (Cornell University)|Jun 22, 2019

Logic, programming, and type systems4 citations

TL;DR

This dissertation develops foundational tools in synthetic homotopy theory using type theory, focusing on classifying types via higher inductive types, descent for homotopy colimits, and reflective subuniverses. It establishes that localization at maps between compact types yields a reflective subuniverse, proving that the subuniverse of N-null types forms a modality—key for constructing modalities and equifibrant replacements in homotopy type theory.

ABSTRACT

The study of homotopy theoretic phenomena in the language of type theory is sometimes loosely called `synthetic homotopy theory'. Homotopy theory in type theory is only one of the many aspects of homotopy type theory, which also includes the study of the set theoretic semantics (models of homotopy type theory and univalence in a meta-theory of sets or categories), type theoretic semantics (internal models of homotopy type theory), and computational semantics, as well as the study of various questions in the internal language of homotopy type theory which are not necessarily motivated by homotopy theory, or questions related to the development of formalized libraries of mathematics based on homotopy type theory. This thesis concerns the development of synthetic homotopy theory.

Motivation & Objective

• To develop synthetic homotopy theory within type theory, focusing on higher inductive types and descent properties.
• To characterize the structure of reflective subuniverses and modalities in homotopy type theory.
• To establish a localization theory at maps between compact types, enabling the construction of modalities.
• To define and analyze the equifibrant replacement operation as a tool for handling homotopy colimits.
• To prove that the subuniverse of N-null types is a modality, providing a new class of modalities in type theory.

Proposed method

• Uses higher inductive types (HITs) such as homotopy pushouts, coequalizers, and sequential colimits to construct types synthetically.
• Applies the descent property for pushouts and reflexive coequalizers to characterize type families over colimits.
• Introduces the join construction to define propositional truncation and the image of a map in type theory.
• Constructs the reflective factorization system for reflexive coequalizers and defines ∆-étale maps to model equifibrant replacement.
• Defines the quasi F-local extension as a pushout construction to initiate localization at families of maps between compact types.
• Uses sequential colimits of quasi F-local extensions to construct the full localization functor, proving its reflectivity.

Experimental results

Research questions

• RQ1 How can homotopy colimits such as pushouts and coequalizers be characterized in type theory using higher inductive types?
• RQ2 What conditions ensure that a subuniverse of types closed under certain maps is reflective?
• RQ3 Can localization at maps between compact types be constructed synthetically in type theory, and does it yield a modality?
• RQ4 How does the equifibrant replacement operation relate to modal descent and reflective factorization systems?
• RQ5 What is the role of the object classifier in classifying types and supporting univalence in synthetic homotopy theory?

Key findings

• The subuniverse of N-null types is shown to be a modality, providing a new class of modalities in homotopy type theory.
• Localization at maps between compact types yields a reflective subuniverse, with the localization functor constructed as a sequential colimit of quasi F-local extensions.
• The initial quasi F-local extension is characterized via a universal property involving diagonal fillers, establishing its role in the localization process.
• The descent property for pushouts and reflexive coequalizers allows type families over colimits to be reconstructed from data on the diagram.
• The equifibrant replacement operation is constructed via a reflective factorization system, generalizing to various homotopy colimits including sequential colimits.
• The proof that the localization functor is reflective relies on the 3-for-2 property and the contractibility of diagonal filler types when the target is F-local.