Virtual double categories of split two-sided 2-fibrations

This paper introduces and studies split two-sided 2-fibrations and locally discrete split two-sided 2-fibrations, using a formal categorical approach. We generalise Street’s notion of split two-sided fibration internal to a 2-category to one internal to a sesquicategory. Given a sesquicategory we construct a virtual double category whose horizontal (loose) morphisms are its internal split two-sided fibrations. Specialising to the sesquicategory of lax natural transformations we obtain the virtual double category of split two-sided 2-fibrations, which we study in detail. We then restrict to the sub-virtual double category of locally discrete split two-sided 2-fibrations and show that therein the usual Yoneda 2-functors satisfy a double-categorical formal notion of Yoneda morphism, which formally captures universal properties similar to those satisfied by the morphisms comprising a Yoneda structure on a 2-category. As a consequence we obtain a ’two-sided Grothendieck correspondence’ of locally discrete split two-sided 2-fibrations $A \nrightarrow B$ and 2-functors $B \to Cat^{A^{op}}$. Restricting to $A = 1$, the terminal 2-category, we improve Buckley and Lambert’s ‘Grothendieck correspondence’ for locally discrete split op-2-fibrations by extending the sense in which it is functorial.

💡 Research Summary

The paper develops a new categorical framework for “split two‑sided 2‑fibrations” by moving the internal‑fibration construction from the setting of 2‑categories to the more flexible setting of sesquicategories. A sesquicategory is a 2‑dimensional structure that has objects, 1‑cells and 2‑cells, but does not require the middle‑four interchange law; consequently lax natural transformations, which fail to compose horizontally, can be accommodated as genuine 2‑cells.

The authors first recall the definition of a sesquicategory and give two key examples, most importantly the sesquicategory 2Catₗₐₓ whose objects are small 2‑categories, whose 1‑cells are 2‑functors, and whose 2‑cells are lax natural transformations. They then introduce the notion of a comma object inside a sesquicategory, generalising Street’s 2‑cotensor construction. Using these comma objects, an internal split two‑sided fibration over a sesquicategory S can be described as a profunctor \