A characterization is given of those bicategories which are biequivalent to bicategories of modules for some suitable base. These bicategories are the correct (non elementary) notion of cosmos, which is shown to be closed under several basic constructions.

We give the definitions of model bicategory and q-homotopy, which are natural generalizations notions category homotopy to context bicategories. For any C, denote by Cfc full sub-bicategory fibrant-cofibrant objects. prove that 2-dimensional localization C at weak equivalences can be computed as a Ho(C) whose objects arrows those 2-cells classes q-homotopies up an equivalence relation. When con...

This work contributes to clarifying several relationships between certain higher categorical structures and the homotopy type of their classifying spaces. Bicategories (in particular monoidal categories) have well understood simple geometric realizations, and we here deal with homotopy types represented by lax diagrams of bicategories, that is, lax functors to the tricategory of bicategories. I...

Abstract. We examine the periodic table of weak n-categories for the lowdimensional cases. It is widely understood that degenerate categories give rise to monoids, doubly degenerate bicategories to commutative monoids, and degenerate bicategories to monoidal categories; however, to understand this correspondence fully we examine the totalities of such structures together with maps between them ...

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 rewriting techniques based on Squier’s Theorem allow us to conclude. In the case of pseudonatural transformations this approach only proves the coherence of part ...

Abstract. 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 isomorphism is subject to a coherence law. We give several examples of compact closed bicategories, then review previous work. In particular, Day a...

Linear bicategories are a generalization of bicategories, in which the one horizontal composition is replaced by two (linked) horizontal compositions. These compositions provide a semantic model for the tensor and par of linear logic: in particular, as composition is fundamentally noncommutative, they provide a suggestive source of models for noncommutative linear logic. In a linear bicategory,...

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 strictfication results of independent interest for symmetric monoidal bicategories and for diagrams of 2-categories. CONTENTS