Double Fibrations

Abstract

This paper defines double fibrations (fibrations of double categories) and describes their key examples and properties. In particular, it shows how double fibrations relate to existing fibrational notions such as monoidal fibrations and discrete double fibrations, proves a representation theorem for double fibrations, and shows how double fibrations are a type of internal fibration.

Descent for internal multicategory functors

“…As stated in the Introduction, we have not considered the 2-dimensional structure of H Lax -T -Alg. However, inspired by the fact that (T, V)-Cat → Set and Cat(T, V) → V for suitable V are fibrations, it might be interesting to explore whether H Lax -T -Alg → D is a double fibration (see [14]), as well as possible applications, even in the more general context where D is a virtual double category. 11.4.…”