nLab
dense subcategory

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 $c\to \overline{c}$ 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 $N:C\to [S^{\mathrm{op}},\mathrm{Set}]$ of objects in $C$ with values in the presheaves on $S$. See nerve and nerve and realization for more on this.

Related concepts

• dense functor

References

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