Modalities in homotopy type theory

[Bas Spitters]

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

Univalent homotopy type theory (HoTT) may be seen as a language for
the category of ∞-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.

https://arxiv.org/abs/1706.07526