dense functor


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 nerve functor (or “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.

Later F. 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.

Also, in arXiv v4 of HTT this notion (for (∞,1)-categories) is referred to as strongly generating, but that term actually 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.


Revised on July 31, 2014 09:53:16 by Thomas Holder (