Bifibrations of Polycategories and Classical Linear Logic

Blanco, Nicolás; Zeilberger, Noam

doi:10.1016/j.entcs.2020.09.003

Cited by 4 publications

(11 citation statements)

References 51 publications

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“…Clearly (i)⇒(ii), while (ii)⇔(iii) follows from §3. The remaining direction (ii)⇒(i) is similar to the universal characterization of * -autonomous categories in [BZ20]. By × Θ, Γ, or ˙∆ we mean the result of combining Table 2.…”

Section: Unifying Universalitymentioning

confidence: 91%

“…We will see some more examples of lnl polycategories in §3, but first we define the basic universal properties that appear therein. Inspired by [BZ20], we say that a morphism ψ in an lnl polycategory containing an object R (linear or nonlinear) in its domain or codomain is universal in R if composing along R induces bijections on homsets of all possible types. For the five possible combination of types for ψ and R, this specializes to the following.…”

Section: Lnl Polycategoriesmentioning

confidence: 99%

“…• For cartesian multicategories, cartesian morphisms specialize to the cartesian and opcartesian morphisms of [LSR17]. • For symmetric polycategories, cartesian morphisms specialize to the cartesian and opcartesian morphisms of [BZ20]. • For categories, cartesian morphisms specialize to the traditional notion of cartesian and opcartesian morphism.…”

Section: Unifying Universalitymentioning

confidence: 99%

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LNL polycategories and doctrines of linear logic

Shulman¹

2021

Preprint

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We define and study LNL polycategories, which abstract the judgmental structure of classical linear logic with exponentials. Many existing structures can be represented as LNL polycategories, including LNL adjunctions, linear exponential comonads, LNL multicategories, IL-indexed categories, linearly distributive categories with storage, commutative and strong monads, CBPV-structures, models of polarized calculi, skew multicategories, as well as ordinary cartesian and symmetric multicategories and monoidal categories, polycategories, and linearly distributive and * -autonomous categories.To study such classes of structures uniformly, we define a notion of LNL doctrine, such that each of these classes of structures can be identified with the algebras for some such doctrine. We show that free algebras for LNL doctrines can be presented by a sequent calculus, and that every morphism of doctrines induces an adjunction between their 2-categories of algebras.

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Section: Unifying Universalitymentioning

confidence: 91%

Section: Lnl Polycategoriesmentioning

confidence: 99%

Section: Unifying Universalitymentioning

confidence: 99%

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LNL polycategories and doctrines of linear logic

Shulman¹

2021

Preprint

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“…We will see some more examples of lnl polycategories in Section 3, but first we define the basic universal properties that appear therein. Inspired by [BZ20], we say that a morphism ψ in an lnl polycategory containing an object R (linear or nonlinear) in its domain or codomain is universal in R if composing along R induces bijections on homsets of all possible types. For the five possible combination of types for ψ and R, this specializes to the following.…”

Section: Lnl Polycategoriesmentioning

confidence: 99%

LNL polycategories and doctrines of linear logic

Shulman¹

2023

Logical Methods in Computer Science

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We define and study LNL polycategories, which abstract the judgmental structure of classical linear logic with exponentials. Many existing structures can be represented as LNL polycategories, including LNL adjunctions, linear exponential comonads, LNL multicategories, IL-indexed categories, linearly distributive categories with storage, commutative and strong monads, CBPV-structures, models of polarized calculi, Freyd-categories, and skew multicategories, as well as ordinary cartesian, symmetric, and planar multicategories and monoidal categories, symmetric polycategories, and linearly distributive and *-autonomous categories. To study such classes of structures uniformly, we define a notion of LNL doctrine, such that each of these classes of structures can be identified with the algebras for some such doctrine. We show that free algebras for LNL doctrines can be presented by a sequent calculus, and that every morphism of doctrines induces an adjunction between their 2-categories of algebras.

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“…• Higher Field Theories. The application of modern category theory to theoretical physics has led to the development of higher gauge theories [BH11], higher algebras in supergravity [Rav21], homotopical extensions of quantum field theories [Sch21], homotopical pregeometric theories [AG21; AGE21], higher topos theory [Lur09], polycategory-based theories [BZ20] and operad-based theories [BSW21]. Although, as we pointed out in 4.4, most of these higher categorical notions are still firmly based on binary morphisms, they represent a general trend towards higher order concepts.…”

Section: Biologymentioning

confidence: 99%

An Invitation to Higher Arity Science

Zapata-Carratalá¹,

Arsiwalla²

2022

Preprint

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Analytical thinking is dominated by binary ideas. From pair-wise interactions, to algebraic operations, to compositions of processes, to network models, binary structures are deeply ingrained in the fabric of most current scientific paradigms. In this article we introduce arity as the generic conceptualization of the order of an interaction between a discrete collection of entities and argue that there is a rich universe of higher arity ideas beyond binarity waiting to be explored. To illustrate this we discuss several higher order phenomena appearing in a wide range of research areas, paying special attention to instances of ternary interactions. From the point of view of formal sciences and mathematics, higher arity thinking opens up new paradigms of algebra, symbolic calculus and logic. In particular, we delve into the special case of ternary structures, as that itself reveals ample surprises: new notions of associativity (or lack thereof) in ternary operations of cubic matrices, ternary isomorphisms and ternary relations, the integration problem of 3-Lie algebras, and generalizations of adjacency in 3-uniform hypergraphs. All these are open problems that strongly suggest the need to develop new ternary mathematics. Finally, we comment on potential future research directions and remark on the transdisciplinary nature of higher arity science.

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Bifibrations of Polycategories and Classical Linear Logic

Cited by 4 publications

References 51 publications

LNL polycategories and doctrines of linear logic

LNL polycategories and doctrines of linear logic

LNL polycategories and doctrines of linear logic

An Invitation to Higher Arity Science

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