Localization and Sheaf Reflectors

Given a triple <!-- MATH $(S,\eta ,\mu )$ --> on a category <!-- MATH $\mathcal{A}$ --> with equalizers, one can form a new triple whose functor is the equalizer of and . Fakir has studied conditions for to be idempotent, that is, to determine a reflective subcategory of <!-- MATH $\mathcal{A}$ --> . Here we regard 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 <!-- MATH $\mathcal{A}$ --> from an injective object , taking <!-- MATH $S = {I^{{I^{( - )}}}}$ --> . We show that this reflector preserves finite limits and that the sheaf reflector for a topology in <!-- MATH $\mathcal{A}$ --> can be obtained in this way. We also show that sheaf reflectors in functor categories can be obtained from a triple of the form <!-- MATH $S = {I^{( - ,I)}},I$ --> 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.