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Indexed type theories

Published online by Cambridge University Press:  22 May 2020

Valery Isaev Open the ORCID record for Valery Isaev [Opens in a new window]
Valery Isaev*
Affiliation:
JetBrains Research and National Research University Higher School of Economics, St. Petersburg, Russia
*
Email: valery.isaev@gmail.com
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Abstract

In this paper, we define indexed type theories which are related to indexed (∞-)categories in the same way as (homotopy) type theories are related to (∞-)categories. We define several standard constructions for such theories including finite (co)limits, arbitrary (co)products, exponents, object classifiers, and orthogonal factorization systems. We also prove that these constructions are equivalent to their type theoretic counterparts such as Σ-types, unit types, identity types, finite higher inductive types, Π-types, univalent universes, and higher modalities.

Keywords

Homotopy type theorycategory theory
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 31 , Issue 1 , January 2021 , pp. 3 - 63
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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