nLab
dense subcategory

Contents

Definition

There are two different notions of dense subcategory DD of a given category CC:

  1. A subcategory DCD\subset C is dense if every object in CC is canonically a colimit of objects in DD.

    This is equivalent to saying that the inclusion functor DCD\hookrightarrow C is a dense functor.

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

  2. A subcategory DCD\subset C is dense if every object cc of CC has a DD-expansion, that is a morphism cc¯c\to\bar{c} of pro-objects in DD which is universal (initial) among all morphisms of pro-objects in DD with domain cc.

    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 May 29, 2013 18:02:50 by Urs Schreiber (89.204.155.181)