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

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)

Last revised on September 30, 2019 at 04:16:13.
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