## Egbert Rijke ; Michael Shulman ; Bas Spitters - Modalities in homotopy type theory

lmcs:3826 - Logical Methods in Computer Science, January 8, 2020, Volume 16, Issue 1 - https://doi.org/10.23638/LMCS-16(1:2)2020
Modalities in homotopy type theory

Authors: Egbert Rijke ; Michael Shulman ; Bas Spitters

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.

Volume: Volume 16, Issue 1
Published on: January 8, 2020
Accepted on: January 8, 2020
Submitted on: August 1, 2017
Keywords: Mathematics - Category Theory,Computer Science - Logic in Computer Science,Mathematics - Logic,F.3.1, F.4.1,F.3.1,F.4.1