Lax pullback complements in partial morphism categories (Q6558472)

The concept of pullback complement was introduced in [\textit{W. Tholen}, Suppl. Rend. Circ. Mat. Palermo (2) 12, 133--138 (1986; Zbl 0616.18001)], while it was established in [\textit{R. Dyckhoff} and \textit{W. Tholen}, J. Pure Appl. Algebra 49, 103--116 (1987; Zbl 0659.18003)] that a pullback complement of every morphism along a given morphism \(s\)\ exists iff \(s\)\ is an exponentiable monomorphism. The concept of lax pullback complement was given in [\textit{S. N. Hosseini} et al., Theory Appl. Categ. 33, 445--475 (2018; Zbl 1408.18006)], being characterized in several ways. This paper aims to characterize the lax pullback complement of a given partial morphism along a total morphism in an \(\mathcal{M}\)-partial morphism category, where \(\mathcal{M}\)\ is an exponentiable stable system.\N\NThe synopsis of the paper goes as follows.\N\N\begin{itemize}\N\item[\S 1] recalls some notions, establishing that certain right adjoints preserve couniversality.\N\N\item[\S 2] establishes that if a lax pullback complement of \(\overrightarrow {u}\)\ along \(s\)\ exists in the partial morphism category \(\mathrm{Par} _{\mathcal{M}}\left( \mathcal{C}\right) \), then a lax pullback complement of \(u\)\ along \(s\)\ exists in \(\mathcal{C}\).\N\N\item[\S 3] shows, with the base category being adhesive [\textit{S. Lack} and \textit{P. Sobociński}, Theor. Inform. Appl. 39, No. 3, 511--545 (2005; Zbl 1078.18010)], that a lax pullback complement of \(\overrightarrow{u}\)\ along an admissible \(s\)\ exists in \(\mathrm{Par}_{\mathcal{M}}\left( \mathcal{C}\right) \), if a lax pullback complement of \(u\)\ along \(s\)\ exists in \(\mathcal{C}\).\N\N\item[\S 4] defines an \(\mathcal{M}\)-cohesive category in such a manner that when the base category \(\mathcal{M}\)-cohesive, the lax pullback complement along arbitrary total maps exist.\N\end{itemize}