A hypervirtual double category is a virtual double category enhanced with additional 2-cells whose vertical target has length 0 (i.e. is a single object rather than a horizontal arrow).

In particular, there is always a “vertical 2-category” consisting of the objects, vertical arrows, and 2-cells whose vertical source and target are both length 0. A virtual double category, by contrast, does not have a vertical 2-category unless it has all units.

Examples

Large (not necessarily locally small) categories, functors, small-set-valued profunctors, and transformations form a hypervirtual double category. Note that only locally small categories have units therein, so the underlying virtual double category does not have enough data to reconstruct the 2-category of large categories, functors, and natural transformations; but the hypervirtual double category does.

Applications

Hypervirtual double categories are a natural context in which to compare proarrow equipments and Yoneda structures, which are two different approaches to “formal category theory”. In both cases there is a notion of “profunctor” which are generally considered to be small-set-valued, but Yoneda structures require non-locally-small categories (the presheaf categories of non-small categories). In this context the Yoneda embedding can be given a universal property relative to the horizontal arrows. See (Koudenburg).