Localization at injectives in complete categories

Abstract:

We consider a complete category $\mathcal {A}$. For each object $I$ of $\mathcal {A}$ we define a functor $Q:\mathcal {A} \to \mathcal {A}$ and obtain a necessary and sufficient condition on $I$ for $Q$, after restricting its codomain, to become a reflector of $\mathcal {A}$ onto the limit closure of $I$. In particular, this condition is satisfied if $I$ is injective in $\mathcal {A}$ with regard to equalizers. Among the special cases of such reflectors are: the reflector onto torsion-free divisible objects associated to an injective $I$ in $\operatorname {Mod} R$; the Samuel compactification of a uniform space; the Stone-Čech compactification. We give a second description of $Q$ in terms of a triple on sets. If $I$ is injective and the functor $Q$ is equivalent to the identity then, under a few extra conditions on $\mathcal {A},{\mathcal {A}^{{\text {op}}}}$ is triplable over sets with regard to the functor taking $A$ to $\mathcal {A}(A,I)$.