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Modal descent

Published online by Cambridge University Press:  26 October 2020

Felix Cherubini Open the ORCID record for Felix Cherubini [Opens in a new window]  and
Egbert Rijke
Felix Cherubini*
Affiliation:
Softwareschneiderei GmbH
Egbert Rijke
Affiliation:
University of Ljubljana
*
*Corresponding author. Email: felix.cherubini@posteo.de
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Abstract

Any modality in homotopy type theory gives rise to an orthogonal factorization system of which the left class is stable under pullbacks. We show that there is a second orthogonal factorization system associated with any modality, of which the left class is the class of ○-equivalences and the right class is the class of ○-étale maps. This factorization system is called the modal reflective factorization system of a modality, and we give a precise characterization of the orthogonal factorization systems that arise as the modal reflective factorization system of a modality. In the special case of the n-truncation, the modal reflective factorization system has a simple description: we show that the n-étale maps are the maps that are right orthogonal to the map $${\rm{1}} \to {\rm{ }}{{\rm{S}}^{n + 1}}$$. We use the ○-étale maps to prove a modal descent theorem: a map with modal fibers into ○X is the same thing as a ○-étale map into a type X. We conclude with an application to real-cohesive homotopy type theory and remark how ○-étale maps relate to the formally etale maps from algebraic geometry.

Keywords

Homotopy type theorymodal homotopy type theorycohesive homotopy type theoryalgebraic geometryfactorization systems

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Type
Paper
Information
Mathematical Structures in Computer Science , Volume 31 , Issue 4 , April 2021 , pp. 363 - 391
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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