Modalities in homotopy type theory

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...

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Bibliographic Details
Published in:	Logical methods in computer science Vol. 16, Issue 1
Main Authors:	Rijke, Egbert, Shulman, Michael, Spitters, Bas
Format:	Journal Article
Language:	English
Published:	Logical Methods in Computer Science e.V 01.01.2020
Subjects:	computer science - logic in computer science f.3.1 f.3.1, f.4.1 f.4.1 mathematics - category theory mathematics - logic
ISSN:	1860-5974, 1860-5974
Online Access:	Get full text
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Summary:	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.
ISSN:	1860-5974 1860-5974
DOI:	10.23638/LMCS-16(1:2)2020