I'm looking for examples of adjunctions involving either the category Gpd of groupoids or the category Cat of small categories.
The insertion of Gpd into Cat has both a left and a right adjoint, so any adjunction involving one of the two can be composed to give a new adjunction involving the other instead.
I know a few easy ones already:
- adjunctions involving the category Set of sets
- taking the free category on a directed graph
- taking the nerve of a category as a simplicial set
As a related question, Cat has a full subcategory SkCat of skeletal categories. Does the insertion of SkCat into Cat admit an adjoint? Obviously we can ask the same for SkGpd.