nLab
dense subcategory

There are two different notions of a dense subcategory $D$ of a category $C$:

1. A subcategory $D\subset C$ is dense if every object in $C$ is canonically a colimit of objects in $D$.

   This is equivalent to saying that the inclusion $D\hookrightarrow C$ 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.

2. A subcategory $D\subset C$ is dense if every object $c$ of $C$ has a $D$-expansion, that is a morphism $c\to \overline{c}$ of pro-objects in $D$ which is universal (initial) among all morphisms of pro-objects in $D$ with source $c$.

   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?.