nLab 2-pro-object

Idea

In the context of 2-category theory, the concept of a 2-pro-object is the categorification of the concept of a pro-object. A 2-pro-object in a 2-category, $\mathcal{C}$, is a 2-functor (or diagram) indexed by a 2-cofiltered 2-category. These 2-pro-objects form a 2-category, $2Pro(\mathcal{C})$, which is closed under small 2-cofiltered pseudolimits.

Pre-composition with the inclusion $c:\mathcal{C} \to 2Pro(\mathcal{C})$ is an equivalence of 2-categories:

$$c^{\ast}: Hom(2Pro(\mathcal{C}),Cat)_+ \to Hom(\mathcal{C},Cat),$$

where $Hom(2Pro(\mathcal{C}),Cat)_+$ is the full subcategory whose objects are those 2-functors that preserve small 2-cofiltered pseudolimits (Descotte & Dubuc, Thrm 2.4.2).

References

• Maria Emilia Descotte, Eduardo Dubuc, A theory of 2-pro-objects (with expanded proofs), (arXiv:1406.5762)

• Maria Emilia Descotte, A theory of 2-pro-objects, a theory of 2-model 2-categories and the 2-model structure for 2-Pro(C), (arXiv:2010.10636)