We characterize abstract Morita contexts in several bicategories. In particular, we use heteromorphisms for the bicategory Prof of categories and profunctors and coreflective subcategories for Cat (categories and functors). In addition, we prove general statements concerning strict Morita contexts, and we give new equivalent forms to the standard notions of adjointness, category equivalence and...

We introduce the notion of weakly globular double categories, a particular class of strict double categories, as a way to model weak 2-categories. We show that this model is suitably equivalent to bicategories and give an explicit description of the functors involved in this biequivalence. As an application we show that groupoidal weakly globular double categories model homotopy 2-types.

Presentations for unbraided, braided and symmetric pseudomonoids are defined. Biequivalences characterising the semistrict bicategories generated by these presentations are proven. It is shown that these biequivalences categorify results in the theory of monoids and commutative monoids, and are generalisations of standard coherence theorems for braided and symmetric monoidal categories.

Kleisli bicategories are a natural environment in which the combinatorics involved in various notions of algebraic theory can be handled in a uniform way. The setting allows a clear account of comparisons between such notions. Algebraic theories, symmetric operads and nonsymmetric operads are treated as examples.

Abstract We apply the theory of weighted bicategorical colimits to study problem existence and computation such birepresentations finitary bicategories. The main application our results is complete classification simple transitive a bicategory studied previously by Zimmermann. confirms conjecture he has made.

We introduce a self-dual, noncommutative, and noncocommutative Hopf algebra HGT which takes for certain Hopf categories (and therefore braided monoidal bicategories) a similar role as the GrothendieckTeichmüller group for quasitensor categories. We also give a result which highly restricts the possibility for similar structures for higher weak n-categories (n ≥ 3) by showing that these structur...