# Reflection of an object of a category

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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).