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Homotopy type-theoretic interpretations of constructive set theories

Published online by Cambridge University Press:  29 November 2019

Cesare Gallozzi Open the ORCID record for Cesare Gallozzi [Opens in a new window]
Cesare Gallozzi*
Affiliation:
School of Mathematics, University of Leeds, Leeds, West Yorkshire, UK, Email: cesaregallozzi@gmail.com
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Abstract

We introduce a family of (k, h)-interpretations for 2 ≤ k ≤ ∞ and 1 ≤ h ≤ ∞ of constructive set theory into type theory, in which sets and formulas are interpreted as types of homotopy level k and h, respectively. Depending on the values of the parameters k and h, we are able to interpret different theories, like Aczel’s CZF and Myhill’s CST. We also define a proposition-as-hproposition interpretation in the context of logic-enriched type theories. The rest of the paper is devoted to characterising and analysing the interpretations considered. The formulas valid in the prop-as-hprop interpretation are characterised in terms of the axiom of unique choice. We also analyse the interpretations of CST into homotopy type theory, providing a comparative analysis with Aczel’s interpretation. This is done by formulating in a logic-enriched type theory the key principles used in the proofs of the two interpretations. Finally, we characterise a class of sentences valid in the (k, ∞)-interpretations in terms of the ΠΣ axiom of choice.

Keywords

Constructive set theorydependent type theoryhomotopy type theorylogic-enriched type theory
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 31 , Issue 1 , January 2021 , pp. 112 - 143
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Copyright
© Cambridge University Press 2019

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References

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