TY - JOUR

T1 - Modal dependent type theory and dependent right adjoints

AU - Birkedal, Lars

AU - Clouston, Ranald

AU - Mannaa, Bassel

AU - Møgelberg, Rasmus Ejlers

AU - M. Pitts, Andrew

AU - Spitters, Bas

PY - 2019/12

Y1 - 2019/12

N2 - 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.

AB - 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.

KW - Dependent type theory

KW - modal logic

KW - category theory

U2 - 10.1017/S0960129519000197

DO - 10.1017/S0960129519000197

M3 - Journal article

SN - 0960-1295

JO - Mathematical Structures in Computer Science

JF - Mathematical Structures in Computer Science

ER -