Modalities in homotopy type theory

TL;DR

This paper advances homotopy type theory by developing the theory of modalities, factorization systems, and subtoposes, including their construction and semantics, to deepen the foundation of mathematics and higher topos theory.

Contribution

It introduces new constructions of modalities and factorization systems in homotopy type theory using localization higher inductive types, and explores their semantics.

Findings

Construction of ($n$-connected, $n$-truncated) factorization systems.

Development of internal presentations of subtoposes.

Semantic models for the introduced constructions.

Abstract

Univalent homotopy type theory (HoTT) may be seen as a language for the category of $\infty$-groupoids. It is being developed as a new foundation for mathematics and as an internal language for (elementary) higher toposes. We develop the theory of factorization systems, reflective subuniverses, and modalities in homotopy type theory, including their construction using a "localization" higher inductive type. This produces in particular the ($n$-connected, $n$-truncated) factorization system as well as internal presentations of subtoposes, through lex modalities. We also develop the semantics of these constructions.