This paper explores the relationship amongst the various simplicial and pseudo-simplicial objects characteristically associated to any bicategory C. It proves the fact that the geometric realizations of all of these possible candidate ‘nerves of C’ are homotopy equivalent. Any one of these realizations could therefore be taken as the classifying space BC of the bicategory. Its other major resul...

A concise guide to very basic bicategory theory, from the definition of a bicategory to the coherence theorem.

We describe a Cat-valued nerve of bicategories, which associates to every bicategory a simplicial object in Cat, called the 2-nerve. This becomes the object part of a 2-functor N : NHom → [∆,Cat], where NHom is a 2-category whose objects are bicategories and whose 1-cells are normal homomorphisms of bicategories. The 2-functor N is fully faithful and has a left biadjoint, and we characterize it...

In this paper, we provide a notion of $$\infty $$ -bicategories fibred in which call 2-Cartesian fibrations. Our definition is formulated using the language marked biscaled simplicial sets: Those are scaled sets equipped with an additional collection triangles containing 2-simplices, lean triangles, addition to edges all degenerate 1-simplices. We prove existence left proper combinatorial model...

Linear bicategories are a generalization of ordinary bicategories in which there are two horizontal (1-cell) compositions corresponding to the “tensor” and “par” of linear logic. Benabou’s notion of a morphism (lax 2-functor) of bicategories may be generalized to linear bicategories, where they are called linear functors. Unfortunately, as for the bicategorical case, it is not obvious how to or...