# Homotopy Type Theory Modalities in homotopy type theory

Modalities in homotopy type theory. Egbert Rijke, Michael Shulman, Bas Spitters.

arXiv:1706.07526

### 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.