nLab
dense subcategory

Contents

Definition

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

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

    This is equivalent to saying that the inclusion functor DC is a dense functor.

    An older name for a dense subcategory in this sense is an adequate subcategory.

  2. A subcategory DC is dense if every object c of C has a D-expansion, that is a morphism cc¯ of pro-objects in D which is universal (initial) among all morphisms of pro-objects in D with domain 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.

Applications

References

See the relevant section of MacLane’s Categories for the Working Mathematician.

Revised on January 10, 2012 22:30:15 by Todd Trimble (74.88.146.52)