nLab codense functor

Idea

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

Definition

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

$Lim((b \downarrow F) \to A \to B) = b$

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

This notion is dual to the notion of dense functor.

Equivalently, a functor $F$ is codense iff $Id_B$, together with identity natural transformation $Id_F\colon F \to F$, is the pointwise right Kan extension, $Ran_F F$, of $F$ along $F$.

Also, $F$ is codense iff its codensity monad is the identity.

A subcategory is codense if the inclusion functor is codense.

Examples

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

Related concepts

• dense

• dense functor

• codensity monad

References

• 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)