Modal dependent type theory and dependent right adjoints

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  • Lars Birkedal
  • Ranald Clouston, Australian National University
  • ,
  • Bassel Mannaa, EToroX Labs
  • ,
  • Rasmus Ejlers Møgelberg, IT-Universitetet i Kobenhavn
  • ,
  • Andrew M. Pitts, University of Cambridge
  • ,
  • Bas Spitters

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.

Original languageEnglish
JournalMathematical Structures in Computer Science
Volume30
Issue2
Pages (from-to)118-138
Number of pages21
ISSN0960-1295
DOIs
Publication statusPublished - Feb 2020

    Research areas

  • category theory, Dependent type theory, modal logic

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