Modal dependent type theory and dependent right adjoints
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 the...
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Veröffentlicht in: | Mathematical structures in computer science 2020-02, Vol.30 (2), p.118-138 |
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creator | Birkedal, Lars Clouston, Ranald Mannaa, Bassel Ejlers Møgelberg, Rasmus Pitts, Andrew M. Spitters, Bas |
description | 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. |
doi_str_mv | 10.1017/S0960129519000197 |
format | Article |
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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.</description><identifier>ISSN: 0960-1295</identifier><identifier>EISSN: 1469-8072</identifier><identifier>DOI: 10.1017/S0960129519000197</identifier><language>eng</language><publisher>Cambridge: Cambridge University Press</publisher><subject>Adjoints ; Semantics ; Syntax</subject><ispartof>Mathematical structures in computer science, 2020-02, Vol.30 (2), p.118-138</ispartof><rights>Copyright Cambridge University Press Feb 2020</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c273t-f82d65694090615e157f89f46e1ad55d5f2430249792f95c1ddcf1d7fc1d3e9d3</citedby><cites>FETCH-LOGICAL-c273t-f82d65694090615e157f89f46e1ad55d5f2430249792f95c1ddcf1d7fc1d3e9d3</cites><orcidid>0000-0003-0097-6188 ; 0000-0001-7775-3471</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>315,781,785,27925,27926</link.rule.ids></links><search><creatorcontrib>Birkedal, Lars</creatorcontrib><creatorcontrib>Clouston, Ranald</creatorcontrib><creatorcontrib>Mannaa, Bassel</creatorcontrib><creatorcontrib>Ejlers Møgelberg, Rasmus</creatorcontrib><creatorcontrib>Pitts, Andrew M.</creatorcontrib><creatorcontrib>Spitters, Bas</creatorcontrib><title>Modal dependent type theory and dependent right adjoints</title><title>Mathematical structures in computer science</title><description>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. 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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.</abstract><cop>Cambridge</cop><pub>Cambridge University Press</pub><doi>10.1017/S0960129519000197</doi><tpages>21</tpages><orcidid>https://orcid.org/0000-0003-0097-6188</orcidid><orcidid>https://orcid.org/0000-0001-7775-3471</orcidid></addata></record> |
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subjects | Adjoints Semantics Syntax |
title | Modal dependent type theory and dependent right adjoints |
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