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Injective types in univalent mathematics

Published online by Cambridge University Press:  05 January 2021

Martín Hötzel Escardó Open the ORCID record for Martín Hötzel Escardó [Opens in a new window]
Martín Hötzel Escardó*
Affiliation:
School of Computer Science, University of Birmingham, Birmingham, UK
*
Email: m.escardo@cs.bham.ac.uk
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Abstract

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We investigate the injective types and the algebraically injective types in univalent mathematics, both in the absence and in the presence of propositional resizing. Injectivity is defined by the surjectivity of the restriction map along any embedding, and algebraic injectivity is defined by a given section of the restriction map along any embedding. Under propositional resizing axioms, the main results are easy to state: (1) Injectivity is equivalent to the propositional truncation of algebraic injectivity. (2) The algebraically injective types are precisely the retracts of exponential powers of universes. (2a) The algebraically injective sets are precisely the retracts of powersets. (2b) The algebraically injective (n+1)-types are precisely the retracts of exponential powers of universes of n-types. (3) The algebraically injective types are also precisely the retracts of algebras of the partial-map classifier. From (2) it follows that any universe is embedded as a retract of any larger universe. In the absence of propositional resizing, we have similar results that have subtler statements which need to keep track of universe levels rather explicitly, and are applied to get the results that require resizing.

Keywords

Injective typeflabby typeKan extensionpartial-map classifierunivalent mathematicsunivalence axiom
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 31 , Issue 1 , January 2021 , pp. 89 - 111
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Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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