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Framed bicategories are double categories satisfying cerain fibrancy conditions. Many structures naturally organize into framed bicategories, e.g. relations, profunctors, adjoints, open Petri nets, polynomials functors, polynomial comonoids, structured cospans, etc. Symmetric monoidal structures on framed bicategories descend to symmetric monoidal structures on horizontal bicategories. The axioms defining symmetric monoidal double categories are considerably more tractible than those defining symmetric monoidal bicategories. It is thus convenient to study ways of lifting a given bicategory into a framed bicategory along an appropriate category of vertical morphisms. Solutions to the problem of lifting bicategories to double categories have classically being useful in expressing Kelly and Street's mates correspondence and in proving the 2-dimensional Seifert-van Kampen theorem of Brown et. al., amongst many other applications. We consider lifting problems in their full generality.

Globularly generated double categories are minimal solutions to lifting problems of bicategories into double categories along given categories of vertical arrows. Globularly generated double categories form a coreflective sub-2-category of general double categories. This, together with an analysis of the internal structure of globularly generated double categories yields a numerical invariant on general double categories. We call this invariant the length. The length of a double category $C$ measures the complexity of mixed compositions of globular and horizontal identity squares of $C$ and thus provides a measure of complexity for lifting problems of bicategories into $C$. It has long being conjectured by the author that framed bicategories are of length 1.

I will explain recent results on the theory of globularly generated double categories, the length invariant, and the theory of framed bicategories, making use of certain types of indexings and opindexings on decorated bicategories.