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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 category_theory.monad.limits.

structure category_theory.reflective {C : Type u₁} {D : Type u₂} [category_theory.category C] [category_theory.category D] (R : D C) :
Type (max u₁ u₂ v₁ v₂)

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 X in D. More generally this applies to objects essentially in the reflective subcategory, see functor.ess_image.unit_iso.


If A is essentially in the image of a reflective functor i, then η_A is an isomorphism. This gives that the "witness" for A being in the essential image can instead be given as the reflection of A, with the isomorphism as η_A.

(For any B in the reflective subcategory, we automatically have that ε_B is an iso.)


If η_A is an isomorphism, then A is in the essential image of i.

If η_A is a split monomorphism, then A is in the reflective subcategory.


Composition of reflective functors.