category theory

# Contents

## Definition

There are two different notions of dense subcategory $D$ of a given 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 functor $D\hookrightarrow C$ is a dense 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 $c\to\bar{c}$ 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

• A dense functor $S \hookrightarrow C$ into a locally small category $C$ induces a good notion of nerve $N : C \to [S^{op}, Set]$ of objects in $C$ with values in the presheaves on $S$. See nerve and nerve and realization for more on this.

## 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)