dense subcategory




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 DD of a given category CC:

  1. A subcategory DCD\subset C is dense if every object in CC is canonically a colimit of objects in DD.

This is equivalent to saying that the inclusion functor DCD\hookrightarrow C is a dense functor.

An older name for a dense subcategory in this sense is an adequate subcategory.

  1. A subcategory DCD\subset C is dense if every object cc of CC has a DD-expansion, that is a morphism cc¯c\to\bar{c} of pro-objects in DD which is universal (initial) among all morphisms of pro-objects in DD with domain cc.

    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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