Related papers: Bicat is not triequivalent to Gray

We provide a complete description of the category of pseudo-categories (including pseudo-functors, natural and pseudo-natural transformations and pseudo modifications). A pseudo-category is a non strict version of an internal category. It…

Given Gray-categories $P$ and $L$, there is a Gray-category $\mathrm{Tricat}_{\mathrm{ls}}(P,L)$ of locally strict trihomomorphisms with domain $P$ and codomain $L$, tritransformations, trimodifications, and perturbations. If the domain $P$…

We form tricategories and the homomorphisms between them into a bicategory, whose 2-cells are certain degenerate tritransformations. We then enrich this bicategory into an example of a three-dimensional structure called a locally cubical…

We describe a finitary 2-monad on a locally finitely presentable 2-category for which not every pseudoalgebra is equivalent to a strict one. This shows that having rank is not a sufficient condition on a 2-monad for every pseudoalgebra to…

We examine the periodic table of weak n-categories for the low-dimensional cases. It is widely understood that degenerate categories give rise to monoids, doubly degenerate bicategories to commutative monoids, and degenerate bicategories to…

Lack described a Quillen model structure on the category GrayCat of Gray-categories and Gray-functors, for which the weak equivalences are the weak 3-equivalences. In this note, we adapt the technique of Gurski, Johnson, and Osorno to show…

We show how the notion of intercategory encompasses a wide variety of three-dimensional structures from the literature, notably duoidal categories, monoidal double categories, cubical bicategories, double bicategories and Gray categories.…

A Quillen model structure on the category Gray-Cat of Gray-categories is described, for which the weak equivalences are the triequivalences. It is shown to restrict to the full subcategory Gray-Gpd of Gray-groupoids. This is used to provide…

We prove coherence theorems for bicategories, pseudofunctors and pseudonatural transformations. These theorems boil down to proving the coherence of some free $(4,2)$-categories. In the case of bicategories and pseudofunctors, existing…

We define biprops as a generalization of coloured props and of symmetric weak multicategories. These are bicategories whose objects form a free monoid. They are equipped with some structure resembling a symmetric strict tensor product. We…

We construct in a unifying way skew-multicategories and multicategories of double and Gray-categories that we call Gray (skew) multicategories. We study their different versions depending on the types of functors and higher transforms. We…

The results of this thesis allows one to replace calculations in tricategories with equivalent calculations in Gray categories (aka semistrict tricategories). In particular the rewriting calculus for Gray categories as used for example by…

We show that every internal biequivalence in a tricategory T is part of a biadjoint biequivalence. We give two applications of this result, one for transporting monoidal structures and one for equipping a monoidal bicategory with invertible…

A compact closed bicategory is a symmetric monoidal bicategory where every object is equipped with a weak dual. The unit and counit satisfy the usual "zig-zag" identities of a compact closed category only up to natural isomorphism, and the…

We continue the project begun in ``The periodic table of $n$-categories for low dimensions I'' by examining degenerate tricategories and comparing them with the structures predicted by the Periodic table. For triply degenerate tricategories…

We generalise the usual notion of fibred category; first to fibred 2-categories and then to fibred bicategories. Fibred 2-categories correspond to 2-functors from a 2-category into 2-Cat. Fibred bicategories correspond to trihomomorphisms…

This is a first of a series of two papers. Our motive is to tackle the question raised in B\"ohm's "The Gray Monoidal Product of Double Categories" from Applied Categorical Structures: which would be an alternative notion to intercategories…

We define a bicategory in which the 0-cells are the entwinings over variable rings. The 1-cells are triples of a bimodule and two maps of bimodules which satisfy an additional hexagon, two pentagons and two (co)unit triangles; and the…

We give the definitions of model bicategory and $q$-homotopy, which are natural generalizations of the notions of model category and homotopy to the context of bicategories. For any model bicategory $\mathcal{C}$, denote by…

In this paper, we address the construction of homotopy bicategories of $(\infty,2)$-categories, which we take as being modeled by 2-fold Segal spaces. Our main result is the concrete construction of a functor $h_2$ from the category of…