Contents

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

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.

## 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.

• 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.