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On generalized algebraic theories and categories with families

Published online by Cambridge University Press:  18 October 2021

Marc Bezem ,
Thierry Coquand Open the ORCID record for Thierry Coquand [Opens in a new window] ,
Peter Dybjer Open the ORCID record for Peter Dybjer [Opens in a new window]  and
Martín Escardó Open the ORCID record for Martín Escardó [Opens in a new window]
Marc Bezem
Affiliation:
Department of Informatics, University of Bergen, Norway
Thierry Coquand
Affiliation:
Department of Computer Science and Engineering, University of Göteborg, Sweden
Peter Dybjer*
Affiliation:
Department of Computer Science and Engineering, Chalmers University of Technology, Sweden
Martín Escardó
Affiliation:
School of Computer Science, University of Birmingham, UK
*
*Corresponding author. Email: [email protected]
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Abstract

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We give a syntax independent formulation of finitely presented generalized algebraic theories as initial objects in categories of categories with families (cwfs) with extra structure. To this end, we simultaneously define the notion of a presentation Σ of a generalized algebraic theory and the associated category CwFΣ of small cwfs with a Σ-structure and cwf-morphisms that preserve Σ-structure on the nose. Our definition refers to the purely semantic notion of uniform family of contexts, types, and terms in CwFΣ. Furthermore, we show how to syntactically construct an initial cwf with a Σ-structure. This result can be viewed as a generalization of Birkhoff’s completeness theorem for equational logic. It is obtained by extending Castellan, Clairambault, and Dybjer’s construction of an initial cwf. We provide examples of generalized algebraic theories for monoids, categories, categories with families, and categories with families with extra structure for some type formers of Martin-Löf type theory. The models of these are internal monoids, internal categories, and internal categories with families (with extra structure) in a small category with families. Finally, we show how to extend our definition to some generalized algebraic theories that are not finitely presented, such as the theory of contextual cwfs.

Keywords

Dependent type theorygeneralized algebraic theorycategory with familiesinitial modelinternal categoryMartin-Löf type theory
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 31 , Special Issue 9: In Homage to Martin Hofmann , October 2021 , pp. 1006 - 1023
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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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