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Modal dependent type theory and dependent right adjoints

Published online by Cambridge University Press:  12 December 2019

Lars Birkedal ,
Ranald Clouston ,
Bassel Mannaa Open the ORCID record for Bassel Mannaa [Opens in a new window] ,
Rasmus Ejlers Møgelberg ,
Andrew M. Pitts Open the ORCID record for Andrew M. Pitts [Opens in a new window]  and
Bas Spitters
Lars Birkedal
Affiliation:
Department of Computer Science, Aarhus University, Denmark
Ranald Clouston
Affiliation:
Research School of Computer Science, Australian National University, Australia
Bassel Mannaa*
Affiliation:
eToroX Labs, Denmark
Rasmus Ejlers Møgelberg
Affiliation:
IT University of Copenhagen, Denmark
Andrew M. Pitts
Affiliation:
Department of Computer Science and Technology, University of Cambridge, UK
Bas Spitters
Affiliation:
Department of Computer Science, Aarhus University, Denmark
*
*Corresponding author. Email: bassel.mannaa@gmail.com
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Abstract

In recent years, we have seen several new models of dependent type theory extended with some form of modal necessity operator, including nominal type theory, guarded and clocked type theory and spatial and cohesive type theory. In this paper, we study modal dependent type theory: dependent type theory with an operator satisfying (a dependent version of) the K axiom of modal logic. We investigate both semantics and syntax. For the semantics, we introduce categories with families with a dependent right adjoint (CwDRA) and show that the examples above can be presented as such. Indeed, we show that any category with finite limits and an adjunction of endofunctors give rise to a CwDRA via the local universe construction. For the syntax, we introduce a dependently typed extension of Fitch-style modal λ-calculus, show that it can be interpreted in any CwDRA, and build a term model. We extend the syntax and semantics with universes.

Keywords

Dependent type theorymodal logiccategory theory
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 30 , Issue 2 , February 2020 , pp. 118 - 138
Copyright
© Cambridge University Press 2019

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