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Elementary fibrations of enriched groupoids

Published online by Cambridge University Press:  19 November 2021

Jacopo Emmenegger ,
Fabio Pasquali  and
Giuseppe Rosolini Open the ORCID record for Giuseppe Rosolini [Opens in a new window]
Jacopo Emmenegger
Affiliation:
School of Computer Science, University of Birmingham, Birmingham, UK
Fabio Pasquali
Affiliation:
DIMA, Università di Genova, Genova, Italy
Giuseppe Rosolini*
Affiliation:
DIMA, Università di Genova, Genova, Italy
*
*Corresponding author. Email: rosolini@unige.it
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Abstract

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The present paper aims at stressing the importance of the Hofmann–Streicher groupoid model for Martin Löf Type Theory as a link with the first-order equality and its semantics via adjunctions. The groupoid model was introduced by Martin Hofmann in his Ph.D. thesis and later analysed in collaboration with Thomas Streicher. In this paper, after describing an algebraic weak factorisation system $$\mathsf {L, R}$$ on the category $${\cal C}-{\cal Gpd}$$ of $${\cal C}$$ -enriched groupoids, we prove that its fibration of algebras is elementary (in the sense of Lawvere) and use this fact to produce the factorisation of diagonals for $$\mathsf {L, R}$$ needed to interpret identity types.

Keywords

Elementary fibrationsalgebraic weak factorisation systemgroupoid model
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 31 , Special Issue 9: In Homage to Martin Hofmann , October 2021 , pp. 958 - 978
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Creative Commons
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, provided the original article is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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