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.
Later F. Ulmer considered the concept for more general functors , not only inclusions , and introduced the name dense for them.
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.