Round squares in the category of frames (Q2839880)

A commutative square \(\mathfrak{b}\cdot h=g\cdot \mathfrak{a}\) in the category \textbf{Frm} of frames and their homomorphisms is called \textit{round} if replacing \(\mathfrak{a}, \mathfrak{b}\) with their right adjoints \(\mathfrak{a}_*, \mathfrak{b}_*\) yields a commutative diagram (that is, \(h\cdot \mathfrak{a}_*=\mathfrak{b}_*\cdot g\)), albeit not necessarily in \textbf{Frm}.NEWLINENEWLINEAfter introducing and studying round squares in \textbf{Frm}, the authors consider several examples of round squares and present interesting characterizations of several variants of compact frames (namely, compact, Lindelöf, realcompact and paracompact frames) in terms of round squares. These characterizations are unified with the following general result for \(\gamma\)-compact frames (for any strong mono-functor \(\gamma\) in the category of completely regular frames):NEWLINENEWLINEA frame \(L\) is \(\gamma\)-compact, i.e., \(\gamma_L: \gamma L\to L\) is an isomorphism, if and only if for every frame homomorphism \(h: M\to L\), the square \(\gamma_L\cdot \gamma(h)=h\cdot \gamma_M\) is round.NEWLINENEWLINEThey also characterize complete strong nearness frames in terms of round squares: a strong nearness frame \(L\) is complete if and only if for every uniform homomorphism \(h: M\to L\) out of a strong nearness frame, the square \(\gamma_L\cdot C(h)=h\cdot \gamma_M\) (where \(C(h)\) denotes the lift of \(h\) to the completions of \(M\) and \(L\)) is round.