By a representation of a category ℱ here is meant a functor from ℱ to a category V of modules over a commutative ring R. The question is whether there is a groupoid G whose category [G,V] of representations is equivalent to the category [ℱ,V] of representations of the given category ℱ. That is to say, is there a groupoid G such that the free V - category RG on G is Morita V - equivalent to the free V - category Rℱ on ℱ ? The groupoid G could be the core groupoid ℱinv of ℱ; that is, the subcategory of ℱ with the same objects but with only the invertible morphisms. Motivating examples come from Dold-Kan-type theorems and a theorem of Nicholas Kuhn [see “Generic representation theory of finite fields in nondescribing characteristic”, Advances in Math 272 (2015) 598–610]. The plan is to describe structure on ℱ which leads to such a result, and includes these and other examples.