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An interpretation of dependent type theory in a model category of locally cartesian closed categories

Published online by Cambridge University Press:  20 September 2021

Martin E. Bidlingmaier Open the ORCID record for Martin E. Bidlingmaier [Opens in a new window]
Martin E. Bidlingmaier*
Affiliation:
Department of Computer Science, Aarhus University, Aarhus, Denmark
*
Email: mbidlingmaier@cs.au.dk
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Abstract

Locally cartesian closed (lcc) categories are natural categorical models of extensional dependent type theory. This paper introduces the “gros” semantics in the category of lcc categories: Instead of constructing an interpretation in a given individual lcc category, we show that also the category of all lcc categories can be endowed with the structure of a model of dependent type theory. The original interpretation in an individual lcc category can then be recovered by slicing. As in the original interpretation, we face the issue of coherence: Categorical structure is usually preserved by functors only up to isomorphism, whereas syntactic substitution commutes strictly with all type-theoretic structures. Our solution involves a suitable presentation of the higher category of lcc categories as model category. To that end, we construct a model category of lcc sketches, from which we obtain by the formalism of algebraically (co)fibrant objects model categories of strict lcc categories and then algebraically cofibrant strict lcc categories. The latter is our model of dependent type theory.

Keywords

Dependent type theorycategorical semanticscoherence
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 31 , Issue 5 , May 2021 , pp. 469 - 494
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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