nLab 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.


In category theory

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.

In shape theory

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


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


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


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


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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