Reflective functors #
Basic properties of reflective functors, especially those relating to their essential image.
Note properties of reflective functors relating to limits and colimits are included in
- to_is_right_adjoint : category_theory.is_right_adjoint R
- to_full : category_theory.full R
- to_faithful : category_theory.faithful R
A functor is reflective, or a reflective inclusion, if it is fully faithful and right adjoint.
For a reflective functor
i (with left adjoint
L), with unit
η, we have
η_iL = iL η.
When restricted to objects in
D given by
i : D ⥤ C, the unit is an isomorphism. In other words,
η_iX is an isomorphism for any
More generally this applies to objects essentially in the reflective subcategory, see
A is essentially in the image of a reflective functor
η_A is an isomorphism.
This gives that the "witness" for
A being in the essential image can instead be given as the
A, with the isomorphism as
B in the reflective subcategory, we automatically have that
ε_B is an iso.)
η_A is an isomorphism, then
A is in the essential image of
η_A is a split monomorphism, then
A is in the reflective subcategory.
Composition of reflective functors.