In topology, a not necessarily continuous function between Hausdorff spaces is dominant, or dense, in the sense that the image of is dense in , precisely if every continuous map to some other Hausdorff space is uniquely determined by .
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 of category is dense if every object of is a colimit of a diagram of objects in , in a canonical way.
A functor is dense if it satisfies the following equivalent conditions.
every object of is the vertex of the following colimit over the comma category :
every object of is the -weighted colimit of . This version generalizes more readily to the enriched context.
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 , not only inclusions , 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.
Let be a category of algebras and such that has a presentation with operations of at most arity . Let be the free -algebra on generators. Then the full subcategory with object is dense in .
In , 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 is dense in this second sense, if every object in admits a -expansion.
A -expansion of an object in is a morphism in such that is in and is the rudimentary system (constant inverse system) corresponding to ; moreover one asks that the morphism is universal among all such morphisms .
Given a dense subcategory one defines an abstract shape category which has the same objects as , but the morphisms are the equivalence classes of morphisms in of -expansions.
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.