There are two different notions of a dense subcategory of a category :
A subcategory is dense if every object in is canonically a colimit of objects in .
This is equivalent to saying that the inclusion is a dense functor. See also the relevant section of Categories Work and the entries on nerve, geometric realization and singular functor?. An older name for a dense subcategory in this sense is an adequate subcategory.
A subcategory is dense if every object of has a -expansion, that is a morphism of pro-objects in which is universal (initial) among all morphisms of pro-objects in with source .
This second notion is used in shape theory. An alternative name for this is a pro-reflective subcategory?, that is a subcategory for which the inclusion has a proadjoint?.