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.
There are actually two different notions of dense subcategory $D$ of a given category $C$:
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.
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.
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.
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