dense functor




In topology, a not necessarily continuous function f:XYf:X\to Y between Hausdorff spaces is dominant, or dense, in the sense that the image of ff is dense in YY, precisely if every continuous map g:YZg:Y\to Z to some other Hausdorff space ZZ is uniquely determined by gfg\circ f.

The concept of a dense functor is a generalization of this concept to functors.

An important special case that was also historically the source of the concept, is the case of a dense subcategory inclusion: a subcategory SS of category CC is dense if every object cc of CC is a colimit of a diagram of objects in SS, in a canonical way.


A functor i:SCi:S\to C is dense if it satisfies the following equivalent conditions.

  1. every object cc of CC is the vertex of the following colimit over the comma category (i/c)(i/c):

    colim((i/c)pr SSiC) \mathrm{colim}((i/c)\stackrel{\mathrm{pr}_S}{\longrightarrow} S \stackrel{i}{\to} C)
  2. every object cc of CC is the C(i,c)C(i-,c)-weighted colimit of ii. This version generalizes more readily to the enriched context.

  3. the corresponding restricted Yoneda embedding C[S op,Set]C \to [S^{op},Set] is fully faithful.

Terminology and History

John Isbell introduced dense subcategories in a seminal paper (Isbell 1960) under the name left adequate. The dual notion of right adequate was also introduced and subcategories satisfying both were called adequate. It was also shown that while the relation of being left (or right) adequate is not transitive, being adequate is transitive. He also brought out interesting connections with set theory and measurable cardinals.

Later in the mid 60s, Friedrich Ulmer considered the concept for more general functors F:CDF:C\to D, not only inclusions I:CDI:C\hookrightarrow D, and introduced the name dense for them.

Independently, Pierre Gabriel worked on this concept and their work coalesced to what was to become the concept of a locally presentable category of their 1971 monograph. It is also good to keep in mind the ‘Abelian’ subcontext in the background, in particular the developments in module theory e.g. Lazard’s (1964) characterization of flat modules as filtered colimits of finitely generated free modules.

More recently, Jacob Lurie has referred to the analogue notion for (∞,1)-categories as strongly generating in a version (arXiv v4) of his HTT, but that term normally means something different.


  1. Let VV be a category of algebras and nn \in \mathbb{N} such that VV has a presentation with operations of at most arity nn. Let vv be the free VV-algebra on nn generators. Then the full subcategory with object vv is dense in VV.

  2. In Set\Set, a singleton space is dense.


There is a different notion of a dense subcategory, often used in shape theory, which has a bit of the same spirit. A full subcategory DCD\subset C is dense in this second sense, if every object in CC admits a DD-expansion.

A DD-expansion of an object XX in CC is a morphism XXX\to \mathbf{X} in proC\mathrm{pro}C 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}.

Given a dense subcategory DCD\subset C 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.


  • William Lawvere, John Isbell’s Adequate Subcategories, TopCom 11 no.1 2006. (link)

  • Peter Gabriel, Friedrich Ulmer, Lokal präsentierbare Kategorien , LNM 221 Springer Heidelberg 1971. (§ 3, pp.38-44)

  • 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, pdf)

  • 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.

Last revised on October 2, 2019 at 06:41:26. See the history of this page for a list of all contributions to it.