nLab dense subcategory

Idea

The concept of a dense subcategory generalizes the concept of a dense subspace from topology to categories. Roughly speaking, a dense subcategory ‘sees’ enough of the ambient category to control the behavior and properties of the latter.

The concept forms part of a related family of concepts concerned with ‘generating objects’ and has some interesting interaction with set theory and measurable cardinals.

Definition

In category theory

There are actually 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.

1. 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.

In shape theory

Beware that in shape theory a different notion of a “dense subcategory” is in use:

Definition

(D-expansion)
A $D$-expansion of an object $X$ in a category $C$ is a morphism $X\to \mathbf{X}$ in the category $\mathrm{pro}C$ of pro-objects such that $\mathbf{X}$ is in $\mathrm{pro}D$ and $X$ is the rudimentary system (constant inverse system) corresponding to $X$; moreover one asks that the morphism is universal among all such morphisms $X\to\mathbf{Y}$.

Definition

(shape-dense subcategory)
A full subcategory $D\subset C$ is dense in the sense of shape theory, if every object in $C$ admits a $D$-expansion (Def. )

Remark

(abstract shape category) Given a shape-dense subcategory $D\subset C$ (Def. ) one defines an abstract shape category $\mathrm{Sh}(C,D)$ which has the same objects as $C$, but the morphisms are the equivalence classes of morphisms in $\mathrm{pro}D$ of $D$-expansions (Def. ).

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.

Related concepts

• dense

There is also the notion of “dense subsite”, but this is not a special case of a dense subcategory.

Related entries

• dense functor

• codensity monad

• space and quantity

• measurable cardinal

• dense subtopos

Related pages

• William Lawvere, John Isbell’s Adequate Subcategories , TopCom 11 no.1 (2006). (link)

References

• John Isbell, Adequate subcategories , Illinois J. Math. 4 (1960) pp.541-552. MR0175954 (euclid).

• John Isbell, Subobjects, adequacy, completeness and categories of algebras , Rozprawy Mat. 36 (1964) pp.1-32. (toc)(full text as pdf)

• John Isbell, Small adequate subcategories , J. London Math. Soc. 43 (1968) pp.242-246.

• John Isbell, Locally finite small adequate subcategories , JPAA 36 (1985) pp.219-220.

• Max Kelly, Basic Concepts of Enriched Category Theory , Cambridge UP 1982. (Reprinted as TAC reprint no.10 (2005); chapter 5, pp.85-112)

• Saunders Mac Lane, Categories for the Working Mathematician , Springer Heidelberg 1998². (section X.6, pp.245ff, 250)

• Horst Schubert, Kategorien II , Springer Heidelberg 1970. (section 17.2, pp.29ff)

• Friedrich Ulmer, Properties of dense and relative adjoint functors , J. of Algebra 8 (1968) pp.77-95.