Modalities in homotopy type theory

Egbert Rijke; Michael Shulman; Bas Spitters

Modalities in homotopy type theory

Egbert Rijke, Michael Shulman, Bas Spitters

Department of Computer Science

Research output: Other contribution › Net publication - Internet publication › Research › peer-review

Abstract

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.

Original language	English
Publication date	27 Jun 2017
Publication status	Published - 27 Jun 2017

Access to Document

https://arxiv.org/abs/1706.07526Licence: Unspecified

Cite this

@misc{98922907a1f64a45a4580b1afd72849f,

title = "Modalities in homotopy type theory",

abstract = "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.",

keywords = "math.CT, cs.LO, math.LO, 03B15, 03B35, 03B45, 03B70, 03F65, 03G30, 18A32, 18B25, 18C35, 18C50, 18D30, F.3.1; F.4.1",

author = "Egbert Rijke and Michael Shulman and Bas Spitters",

note = "v2: fixed some references Senest {\ae}ndret: 27/06/2017",

year = "2017",

month = jun,

day = "27",

language = "English",

type = "Other",

}

TY - ICOMM

T1 - Modalities in homotopy type theory

AU - Rijke, Egbert

AU - Shulman, Michael

AU - Spitters, Bas

N1 - v2: fixed some references Senest ændret: 27/06/2017

PY - 2017/6/27

Y1 - 2017/6/27

N2 - 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.

AB - 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.

KW - math.CT

KW - cs.LO

KW - math.LO

KW - 03B15, 03B35, 03B45, 03B70, 03F65, 03G30, 18A32, 18B25, 18C35, 18C50, 18D30

KW - F.3.1; F.4.1

M3 - Net publication - Internet publication

ER -