Notions of subcategory
There are two different notions of dense subcategory of a given category :
A subcategory is dense if every object in is canonically a colimit of objects in .
This is equivalent to saying that the inclusion functor is a dense 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 domain .
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.
See the relevant section of MacLane’s Categories for the Working Mathematician.
Revised on May 29, 2013 18:02:50
by Urs Schreiber