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- W2810176807 abstract "We present a type theory dealing with non-linear, ordinary dependent types (which we will call cartesian) and linear types, where both constructs may depend on terms of the former. In the interplay between these, we find new type formers $sqcap_{x:A}B$ and $sqsubset_{x:A}B$, akin to $Pi$ and $Sigma$, but where the dependent type $B$, (and therefore the resulting construct) is a linear type. These can be seen as internalizing universal and existential quantification of linear predicates. We also consider two modalities, $M$ and $L$, transforming linear types into cartesian types and vice versa. The theory is interpreted in a split comprehension category $pi:mathcal{T}tomathcal{C}^to$ accompanied by a split symmetric monoidal fibration, $pi: mathcal{L}tomathcal{C}$. This structure determines, for any context $Gamma$, fibers $mathcal{T}_Gamma$ and $mathcal{L}_Gamma$; the category of cartesian types and the monoidal category of linear types over $Gamma$, respectively. Here, the type formers $sqcap_{x:A}$ and $sqsubset_{x:A}$ are understood as right and left adjoints of the monoidal reindexing functor $pi_A^*:mathcal{L}_Gammatomathcal{L}_{Gamma.A}$. The operators $M$ and $L$ induce a fiberwise adjunction $L dashv M$ between $mathcal{L}$ and $mathcal{T}$, where the traditional exponential modality is understood as the comonad $! = LM$. We provide a model of this theory called the Diagram model, which extends the groupoid model of dependent type theory to accommodate linear types. Here, cartesian types are interpreted as a family of groupoids, while linear types are interpreted as diagrams $A:Gammatomathcal{V}$ in any symmetric monoidal category $mathcal{V}$. We show that the diagrams model can under certain conditions support a linear analogue of the univalence axiom, and provide some discussion on the higher-dimensional nature of linear dependent types." @default.
- W2810176807 created "2018-07-10" @default.
- W2810176807 creator A5081085703 @default.
- W2810176807 date "2018-06-25" @default.
- W2810176807 modified "2023-09-27" @default.
- W2810176807 title "A diagram model of linear dependent type theory" @default.
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- W2810176807 hasPublicationYear "2018" @default.
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