Reflection of an object of a category

reflector of an object of a category

Let $ \mathfrak C $ be a subcategory of a category $ \mathfrak K $; an object $ B \in \mathfrak C $ is called a reflection of an object $ A \in \mathfrak K $ in $ \mathfrak C $, or a $ \mathfrak C $- reflection, if there exists a morphism $ \pi : A \rightarrow B $ such that for any object $ X $ of $ \mathfrak C $ the mapping

$$ H _ {X} ( \pi ) : H _ {\mathfrak C} ( B, X) \rightarrow H _ {\mathfrak K} ( A, X) $$

is bijective. In other words, for any morphism $ \alpha : A \rightarrow X $ there is a unique morphism $ \alpha ^ \prime : B \rightarrow X \in \mathfrak C $ such that $ \alpha = \pi \alpha ^ \prime $. A $ \mathfrak C $- reflection of an object $ A $ is not uniquely defined, but any two $ \mathfrak C $- reflections of an object $ A $ are isomorphic. The $ \mathfrak C $- reflection of an initial object of $ \mathfrak K $ is an initial object in $ \mathfrak C $. The left adjoint of the inclusion functor $ \mathfrak C \rightarrow \mathfrak K $( if it exists), i.e. the functor assigning to an object of $ \mathfrak K $ its reflection in $ \mathfrak C $, is called a reflector.

Examples. In the category of groups the quotient group $ G/G ^ \prime $ of an arbitrary group $ G $ by its commutator subgroup is a reflection of $ G $ in the subcategory of Abelian groups. For an Abelian group $ A $, the quotient group $ A/T( A) $ by its torsion subgroup $ T( A) $ is a reflection of $ A $ in the full subcategory of torsion-free Abelian groups. The injective hull $ \widetilde{A} $ of the group $ A/T( A) $ is a reflection of the groups $ A $ and $ A/T( A) $ in the subcategory of full torsion-free Abelian groups.

Reflections are usually examined in full subcategories. A full subcategory $ \mathfrak C $ of a category $ \mathfrak K $ in which there are reflections for all objects of $ \mathfrak K $ is called reflective (cf. Reflexive category).

Comments

The reflection of an object solves a universal problem (cf. Universal problems).