Modalities in homotopy type theory

Abstract

Univalent homotopy type theory (HoTT) may be seen as a language for thecategory of $\infty$-groupoids. It is being developed as a new foundation formathematics and as an internal language for (elementary) higher toposes. Wedevelop the theory of factorization systems, reflective subuniverses, andmodalities 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 internalpresentations of subtoposes, through lex modalities. We also develop thesemantics of these constructions