nLab codense functor




The concept of a codense functor is the dual of dense functor.


Let F:ABF\colon A \to B be a functor between categories. It is codense when for each bBb \in B the following is true:

Lim((bF)AB)=bLim((b \downarrow F) \to A \to B) = b

(where (bF)(b \downarrow F) is a comma category (in this case the under category) from bb to the functor FF.

This notion is dual to the notion of dense functor.

Equivalently, a functor FF is codense iff Id BId_B, together with identity natural transformation Id F:FFId_F\colon F \to F, is the pointwise Kan extension of FF along FF.

Also, FF is codense iff its codensity monad is the identity.

A subcategory is codense if the inclusion functor is codense.


  • Let II denote the unit interval. Then the full subcategory of the category of compact topological spaces TT whose only object is I 2I^2 is a dense subcategory of TT (Ulmer 68, p.80).


  • Friedrich Ulmer, Properties of Dense and Relative Adjoint Functors, Journal of Algebra 8, 77-95 (1968)

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

Last revised on May 31, 2022 at 12:07:57. See the history of this page for a list of all contributions to it.