Title: Modalities in homotopy type theory,
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.