On reflective-coreflective equivalence and associated pairs (Q2884469)

Given a category \({\mathcal C}\) and two its full subcategories \({\mathcal M}\) and \({\mathcal N}\), the latter (resp. former) being (resp. co)reflective in \({\mathcal C}\), Corollary~4.4 of~\textit{E.~Bedos}, \textit{S.~Kaliszewski} and \textit{J.~Quigg} [``Reflective-coreflective equivalence'', Theory Appl. Categ. 25, 142--179 (2011; Zbl 1232.18002)] showed the necessary and sufficient conditions for the respective adjunction between \({\mathcal M}\) and \({\mathcal N}\) to be an equivalence (called in the paper \textit{adjoint equivalence}). Moreover, motivated by some aspects of the theory of \(C^*\)-algebras (e.g., \(C^*\)-dynamical systems), the authors introduced in Section~5 of the above manuscript additional requirements on an adjoint equivalence, calling the new one \textit{``maximal-normal'' type equivalence}. In particular, they presented the necessary and sufficient conditions on the given categories \({\mathcal C}, {\mathcal M}\) and \({\mathcal N}\), to induce a ``maximal-normal'' type equivalence. The purpose of the present short manuscript is to show that the concept of the authors is nothing else than the notion of \textit{associated pair} of subcategories of a given category, considered in Section~2 of~\textit{G.~M.~Kelly} and \textit{F.~W.~Lawvere} [``On the complete lattice of essential localizations'', Bull. Soc. Math. Belg., Sér. A 41, No. 2, 289--319 (1989; Zbl 0686.18005)], which, in its turn, is based in the concept of \textit{(co)orthogonality} between a morphism and an object of a category of~\textit{P.~J.~Freyd} and \textit{G.~M.~Kelly} [``Categories of continuous functors. I'', J. Pure Appl. Algebra 2, 169--191 (1972; Zbl 0257.18005)]. The result in question is stated in Theorem~2.2 on page 535, providing its proof in full detail.NEWLINENEWLINEThe manuscript is well-written (no typos), self-contained with respect to the needed definitions (the two given proofs, however, rely heavily on the stuff of the above paper of the authors) and easy to follow (no sophisticated techniques).