Skew monoidal categories and skew multicategories

Bourke, John; Lack, Stephen

doi:10.1016/j.jalgebra.2018.02.039

Cited by 16 publications

(28 citation statements)

References 23 publications

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“…Since left universal maps are closed under (scut) composition, a skew multicategory is left representable if and only if it is nullary-binary left representable (cf. Proposition 4.5 of [5]). The following is stated as one of the main results of [5].…”

Section: Definitionmentioning

confidence: 99%

“…Theorem 7 (Bourke & Lack [5]). There is a 2-equivalence between the 2-category of skew monoidal categories and lax monoidal functors and the 2-category of left representable skew multicategories and skew multifunctors (that do not necessarily preserve left universal multimaps).…”

Section: Definitionmentioning

confidence: 99%

“…Szachlányi's paper was immediately noticed by Street, Lack and colleagues who have by now published a whole series of followup works [13,14,6,4,5]. Although the definition of skew monoidal category is simple, it entails some remarkably subtle properties.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Sequent Calculus of Skew Monoidal Categories

Uustalu

Veltri

Zeilberger

2018

Electronic Notes in Theoretical Computer Science

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Szlachányi's skew monoidal categories are a well-motivated variation of monoidal categories in which the unitors and associator are not required to be natural isomorphisms, but merely natural transformations in a particular direction. We present a sequent calculus for skew monoidal categories, building on the recent formulation by one of the authors of a sequent calculus for the Tamari order (skew semigroup categories). In this calculus, antecedents consist of a stoup (an optional formula) followed by a context, and the connectives behave like in the standard monoidal sequent calculus except that the left rules may only be applied in stoup position. We prove that this calculus is sound and complete with respect to existence of maps in the free skew monoidal category, and moreover that it captures equality of maps once a suitable equivalence relation is imposed on derivations. We then identify a subsystem of focused derivations and establish that it contains exactly one canonical representative from each equivalence class. This coherence theorem leads directly to simple procedures for deciding equality of maps in the free skew monoidal category and for enumerating any homset without duplicates. Finally, and in the spirit of Lambek's work, we describe the close connection between this proof-theoretic analysis and Bourke and Lack's recent characterization of skew monoidal categories as left representable skew multicategories. We have formalized this development in the dependently typed programming language Agda.

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Section: Definitionmentioning

confidence: 99%

Section: Definitionmentioning

confidence: 99%

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The Sequent Calculus of Skew Monoidal Categories

Uustalu

Veltri

Zeilberger

2018

Electronic Notes in Theoretical Computer Science

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“…Moreover, this defines a 2-functor from the 2-category of symmetric skew closed categories, symmetric closed functors, and closed natural transformations to the 2-category of symmetric multicategories, symmetric multifunctors, and multinatural transformations. (See [BL17b] for an abstract proof of the non-symmetric version of this statement; see also [Man12] for a more concrete proof of the non-symmetric non-skew version). A and B, let Hom(A, B) denote the pseudo double category whose objects are pseudo double functors A −→ B, whose vertical morphisms are vertical transformations, whose horizontal morphisms are pseudo horizontal transformations, and whose cells are modifications.…”

Section: Appendix a The Multicategory Of Pseudo Double Categoriesmentioning

confidence: 99%

How strict is strictification?

Campbell

2019

Journal of Pure and Applied Algebra

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The subject of this paper is the higher structure of the strictification adjunction, which relates the two fundamental bases of three-dimensional category theory: the Graycategory of 2-categories and the tricategory of bicategories. We show that -far from requiring the full weakness provided by the definitions of tricategory theory -this adjunction can be strictly enriched over the symmetric closed multicategory of bicategories defined by Verity. Moreover, we show that this adjunction underlies an adjunction of bicategory-enriched symmetric multicategories. An appendix introduces the symmetric closed multicategory of pseudo double categories, into which Verity's symmetric multicategory of bicategories embeds fully.

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“…These processes lose structure, but choosing for T an only slighter more complex operad L, we find that colax L-algebras encode the skew monoidal structure entirely. These results are used in our companion paper [2] which introduces and studies skew multicategories, the multicategorical analogue of skew monoidal categories.…”

Section: Introductionmentioning

confidence: 99%

Free skew monoidal categories

Bourke

Lack

2018

Journal of Pure and Applied Algebra

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In the paper Triangulations, orientals, and skew monoidal categories, the free monoidal category Fsk on a single generating object was described. We sharpen this by giving a completely explicit description of Fsk, and so of the free skew monoidal category on any category. As an application we describe adjunctions between the operad for skew monoidal categories and various simpler operads. For a particular such operad L, we identify skew monoidal categories with certain colax L-algebras.

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Skew monoidal categories and skew multicategories

Abstract: We describe a perfect correspondence between skew monoidal categories and certain generalised multicategories, called skew multicategories, that arise in nature.

Cited by 16 publications

References 23 publications

The Sequent Calculus of Skew Monoidal Categories

The Sequent Calculus of Skew Monoidal Categories

How strict is strictification?

Free skew monoidal categories

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