Localization and Sheaf Reflectors

Given a triple $(S,\eta ,\mu )$ on a category $\mathcal {A}$ with equalizers, one can form a new triple whose functor $Q$ is the equalizer of $\eta S$ and $S\eta$. Fakir has studied conditions for $Q$ to be idempotent, that is, to determine a reflective subcategory of $\mathcal {A}$. Here we regard $S$ as the composition of an adjoint pair of functors and give several new such conditions. As one application we construct a reflector in an elementary topos $\mathcal {A}$ from an injective object $I$, taking $S = {I^{{I^{( - )}}}}$. We show that this reflector preserves finite limits and that the sheaf reflector for a topology in $\mathcal {A}$ can be obtained in this way. We also show that sheaf reflectors in functor categories can be obtained from a triple of the form $S = {I^{( - ,I)}},I$ injective, which we studied in a previous paper. We deduce that the opposite of a sheaf subcategory of a functor category is tripleable over Sets.