Nick Gurski

The low-dimensional structures formed by tricategories

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 bicategory, this being a bicategory enriched in the monoidal 2-category of pseudo double categories. Finally, we show that every sufficiently well-behaved locally cubical bicategory gives rise to a tricategory, and thereby deduce the existence of a tricategory of tricategories.

Infinite loop spaces, and coherence for symmetric monoidal bicategories

Abstract This paper proves three different coherence theorems for symmetric monoidal bicategories. First, we show that in a free symmetric monoidal bicategory every diagram of 2-cells commutes. Second, we show that this implies that the free symmetric monoidal bicategory on one object is equivalent, as a symmetric monoidal bicategory, to the discrete symmetric monoidal bicategory given by the disjoint union of the symmetric groups. Third, we show that every symmetric monoidal bicategory is equivalent to a strict one. We give two topological applications of these coherence results. First, we show that the classifying space of a symmetric monoidal bicategory can be equipped with an E ∞ structure. Second, we show that the fundamental 2-groupoid of an E n space, n ≥ 4 , has a symmetric monoidal structure. These calculations also show that the fundamental 2-groupoid of an E 3 space has a sylleptic monoidal structure.

K -theory for 2-categories

Abstract We establish an equivalence of homotopy theories between symmetric monoidal bicategories and connective spectra. For this, we develop the theory of Γ-objects in 2-categories. In the course of the proof we establish strictification results of independent interest for symmetric monoidal bicategories and for diagrams of 2-categories.

The Gray Tensor Product Via Factorisation

We discuss the folklore construction of the Gray tensor product of 2-categories as obtained by factoring the map from the funny tensor product to the cartesian product. We show that this factorisation can be obtained without using a concrete presentation of the Gray tensor product, but merely its defining universal property, and use it to give another proof that the Gray tensor product forms part of a symmetric monoidal structure. The main technical tool is a method of producing new algebra structures over Lawvere 2-theories from old ones via a factorisation system.

Stable Postnikov data of Picard 2–categories

Picard 2-categories are symmetric monoidal 2-categories with invertible 0-, 1-, and 2-cells. The classifying space of a Picard 2-category

Extending homotopy theories across adjunctions

\mathcal{D}

Coherence in Three-Dimensional Category Theory

is an infinite loop space, the zeroth space of the

BIEQUIVALENCES IN TRICATEGORIES

K

Cyclic multicategories, multivariable adjunctions and mates

-theory spectrum

The periodic table of

K\mathcal{D}