The Gray Tensor Product Via Factorisation

Bourke, John; Gurski, Nick

doi:10.1007/s10485-016-9467-6

Cited by 23 publications

(8 citation statements)

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“…In this section we construct an adjunction − ⊗ D ⊣ ⟦D, −⟧ of endofunctors on the category DblCat of double categories, for any double category D. Our line of reasoning is similar to [3,Proposition 3.10]. The occurring double functor ⊗ ∶ DblCat × DblCat → DblCat is our candidate Gray monoidal product on DblCat.…”

Section: Existencementioning

confidence: 99%

The Gray Monoidal Product of Double Categories

Böhm

2019

Appl Categor Struct

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The category of double categories and double functors is equipped with a symmetric closed monoidal structure. For any double category A, the corresponding internal hom functor ⟦A, −⟧ sends a double category B to the double category whose 0-cells are the double functors A → B, whose horizontal and vertical 1-cells are the horizontal and vertical pseudotransformations, respectively, and whose 2cells are the modifications. Some well-known functors of practical significance are checked to be compatible with this monoidal structure.

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Section: Existencementioning

confidence: 99%

The Gray Monoidal Product of Double Categories

Böhm

2019

Appl Categor Struct

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“…It follows from Theorem 5.33 that, for all bicategories B and C, we can define the bicategory of functors, oplax transformations, and modifications between B and C as G[ B, C] s , and the bicategory of functors, pseudo-natural transformations, and modifications as G[ B, C] ps . The latter is part of a monoidal closed structure on BiCat with the better-known "pseudo" version of the Gray product, see for example [BG16], but we cannot see any added insight from the perspective of merge-bicategories.…”

Section: Merge-bicategories and Higher Morphismsmentioning

confidence: 97%

Weak units, universal cells, and coherence via universality for bicategories

Hadzihasanovic¹

2018

Preprint

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Poly-bicategories generalise planar polycategories in the same way as bicategories generalise monoidal categories. In a poly-bicategory, the existence of enough 2-cells satisfying certain universal properties (representability) induces coherent algebraic structure on the 2-graph of single-input, single-output 2-cells. A special case of this theory was used by Hermida to produce a proof of strictification for bicategories. No full strictification is possible for higher-dimensional categories, seemingly due to problems with 2-cells that have degenerate boundaries; it was conjectured by C. Simpson that semi-strictification excluding units may be possible. We study poly-bicategories where 2-cells with degenerate boundaries are barred, and show that we can recover the structure of a bicategory through a different construction of weak units. We prove that the existence of these units is equivalent to the existence of 1-cells satisfying lower-dimensional universal properties, and study the relation between preservation of units and universal cells. Then, we introduce merge-bicategories, a variant of poly-bicategories with more composition operations, which admits a natural monoidal closed structure, giving access to higher morphisms. We derive equivalences between morphisms, transformations, and modifications of representable merge-bicategories and the corresponding notions for bicategories. Finally, we prove a semi-strictification theorem for representable mergebicategories with a choice of composites and units.

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“…For the sake of completeness, we provide in the Appendix (see §B) a purely algebraic description by generators and relations of the Gray tensor product A B of a pair of 2categories A and B. The definition is somewhat involved however, and we thus find more convenient to describe below a characterization of the Gray tensor product A B of two small 2-categories A , B adapted from the work by Bourke and Gurski [6]. Using this specific formulation, we establish the main result of the section, which states that the Gray tensor product of 2-categories A , B → A B preserves coreflexive equalizers componentwise.…”

Section: The Gray Tensor Productmentioning

confidence: 99%

“…using the notations f 1 , f 2 , g 1 , g 2 given in (39). It is possible to define the Gray tensor product A B directly from there, along an idea developed by Bourke and Gurski in [6]. Every small 2-category A comes equipped with an underlying category of objects and morphisms noted |A | and with an underlying set of objects noted ||A ||.…”

Section: A a Concise Characterization Of The Gray Tensor Productmentioning

confidence: 99%