Linking the closure and orthogonality properties of perfect morphisms in a category (Q2713596)

Let \((R,r)\) be a pointed endofunctor of a category \({\mathcal X}\) and let \(r_X : X \rightarrow RX\) be the natural morphism induced by \((R,r)\). A morphism \(f:X\rightarrow Y\) in \({\mathcal X}\) is called \((R,r)\)-perfect, if the diagram below is a pullback diagram: NEWLINE\[NEWLINE\begin{tikzcd} NEWLINEX \ar[r,"f"]\ar[d,"r_x" '] & Y\ar[d,"r_y"]\\ NEWLINERX \ar[r,"Rf" '] & RY . NEWLINE\end{tikzcd} NEWLINE\]NEWLINEThe author introduces also others generalisations of perfect maps coming from characterisations of perfect maps in Tychonov topological spaces. He shows principal interrelations of these introduced notions of perfect morphisms (Theorem 6.1).