Intersection Type Distributors

Olimpieri, Federico

doi:10.1109/lics52264.2021.9470617

Cited by 10 publications

(5 citation statements)

References 40 publications

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“…Bicategories of profunctors are now prominent in the analysis of linear logic and the λ -calculus ( [7,11,22]), and game semantics employs a variety of span-like constructions that compose weakly ( [31,3]). These models have also influenced the development of 2-dimensional type theories ( [8,34]). This paper supports these developments from the perspective of call-by-value languages.…”

Section: Bicategorical Modelsmentioning

confidence: 99%

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Effectful Semantics in 2-Dimensional Categories: Premonoidal and Freyd Bicategories

Paquet,

Saville

2023

Electron. Proc. Theor. Comput. Sci.

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Premonoidal categories and Freyd categories provide an encompassing framework for the semantics of call-by-value programming languages. Premonoidal categories are a weakening of monoidal categories in which the interchange law for the tensor product may not hold, modelling the fact that effectful programs cannot generally be re-ordered. A Freyd category is a pair of categories with the same objects: a premonoidal category of general programs, and a monoidal category of 'effect-free' programs which do admit re-ordering.Certain recent innovations in semantics, however, have produced models which are not categories but bicategories. Here we develop the theory to capture such examples by introducing premonoidal and Freyd structure in a bicategorical setting. The second dimension introduces new subtleties, so we verify our definitions with several examples and a correspondence theorem-between Freyd bicategories and certain actions of monoidal bicategories-which parallels the categorical framework.

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Section: Bicategorical Modelsmentioning

confidence: 99%

“…In these models, the 2-cells play a central role in characterizing the composition operation for morphisms, and additionally provide refined semantic information (see e.g. [17,8,48,34,22]).…”

Section: Introductionmentioning

confidence: 99%

Effectful Semantics in 2-Dimensional Categories: Premonoidal and Freyd Bicategories

Paquet,

Saville

2023

Electron. Proc. Theor. Comput. Sci.

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“…The bicategory Prof Σ is cartesian closed [FGHW08] and forms the basis for a number of models for typed and untyped λ-calculus. These models can be used, for instance, to track the symmetries of resource usage [Oli21] or the combinatorics of reduction paths [TAO17,TAO18].…”

Section: Introductionmentioning

confidence: 99%

Stabilized profunctors and stable species of structures

Fiore,

Galal,

Paquet

2024

Logical Methods in Computer Science

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We introduce a bicategorical model of linear logic which is a novel variation of the bicategory of groupoids, profunctors, and natural transformations. Our model is obtained by endowing groupoids with additional structure, called a kit, to stabilize the profunctors by controlling the freeness of the groupoid action on profunctor elements. The theory of generalized species of structures, based on profunctors, is refined to a new theory of \emph{stable species} of structures between groupoids with Boolean kits. Generalized species are in correspondence with analytic functors between presheaf categories; in our refined model, stable species are shown to be in correspondence with restrictions of analytic functors, which we characterize as being stable, to full subcategories of stabilized presheaves. Our motivating example is the class of finitary polynomial functors between categories of indexed sets, also known as normal functors, that arises from kits enforcing free actions. We show that the bicategory of groupoids with Boolean kits, stable species, and natural transformations is cartesian closed. This makes essential use of the logical structure of Boolean kits and explains the well-known failure of cartesian closure for the bicategory of finitary polynomial functors between categories of set-indexed families and cartesian natural transformations. The paper additionally develops the model of classical linear logic underlying the cartesian closed structure and clarifies the connection to stable domain theory.

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“…The bicategory Prof Σ is cartesian closed and forms the basis for a number of models for typed and untyped λ-calculus. These models can be used, for instance, to track the symmetries of resource usage [39] or the combinatorics of reduction paths [46,47].…”

Section: Introductionmentioning

confidence: 99%

Stabilized profunctors and stable species of structures

Fiore¹,

Galal²,

Paquet³

2023

Preprint

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We introduce a new bicategorical model of linear logic based on profunctors between groupoids. This model is a new variation of the usual bicategory of profunctors, obtained by endowing groupoids with additional structure to constrain the profunctors.One goal of this new model is to provide a formal bridge between the model of finitary polynomial functors, also known as normal functors, and the combinatorial theory of generalized species of structures. Our approach consists in viewing finitary polynomial functors as analytic functors generated by free generalized species.The main conceptual novelty is the notion of kit, designed to control the extent to which species are free. We study kits from both combinatorial and logical perspectives. Profunctors that respect the kit structure are called stabilized, and the bicategory of stabilized profunctors gives rise to stable species of structures, a cartesian closed bicategory that embeds finitary polynomial functors.Stabilized profunctors and stable species can be given an extensional presentation as certain functors between subcategories of presheaves determined by the kit. This gives a strict 2-categorical presentation of the same model.

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Intersection Type Distributors

Cited by 10 publications

References 40 publications

Effectful Semantics in 2-Dimensional Categories: Premonoidal and Freyd Bicategories

Effectful Semantics in 2-Dimensional Categories: Premonoidal and Freyd Bicategories

Stabilized profunctors and stable species of structures

Stabilized profunctors and stable species of structures

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