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category theory

# Contents

## 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.

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

## 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.

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