Quotients of span categories that are allegories and the representation of regular categories (Q2105672)

By identifying vertically isomorphic morphisms in \textit{J. Bénabou}'s [Lect. Notes Math. 47, 1--77 (1967; Zbl 1375.18001)] bicategory \(\mathcal{S}pan(\mathcal{C})\) of spans of morphisms in a category \(\mathcal{C}\) with pullbacks, one obtains the category \[ \mathsf{Span}(\mathcal{C}) \] in which spans are composed horizontally via pullback in \(\mathcal{C}\). If \(\mathcal{C}\) is regular so that \(\mathcal{C}\) has also binary products and a pullback-stable (regular epi, mono)-factorization system, one may similarly form the category \[ \mathsf{Rel}(\mathcal{C}) \] of sets and relations in \(\mathcal{C}\), with the horizontal composite of a composable pair of relations obtained as a regular image of their span composite. which is the prototypical example of a unitary and tabular allegory [\textit{P. T. Johnstone}, Sketches of an elephant. A topos theory compendium. I. Oxford: Clarendon Press (2002; Zbl 1071.18001)]. \textit{D. Pavlović} [J. Pure Appl. Algebra 99, No. 1, 9--34 (1995; Zbl 0829.18002)] considered, without any epi- or mono restrictions, an arbitrary pullback-stable factorization system \((\mathcal{E},\mathcal{M})\) of a category \(\mathcal{C}\) with binary products and pullbacks, forming the category \[ \mathsf{Rel}_{\mathcal{M}}(\mathcal{C}) \] whose morphisms are represented by those spans \((A\leftarrow R\rightarrow B)\) whose induced morphism \(R\rightarrow A\times B\) lies in \(\mathcal{M}\). This paper takes a fresh look at this category by treating it as a quotient category of \(\mathsf{Span}(\mathcal{C})\), The synopsis of the paper goes as follows. \begin{itemize} \item[\S 2] describes, for any pullback-stable class \(\mathcal{E}\) of morphisms in \(\mathcal{C}\) containing all isomorphisms and being closed under composition, a compatible equivalence relation \(\sim_{\mathcal{E}}\) on \(\mathsf{Span}(\mathcal{C})\) which renders its quotient category \[ \mathsf{Span}_{\mathcal{E}}(\mathcal{C}) \] isomorphic to \(\mathsf{Rel}_{\mathcal{M}}(\mathcal{C})\) whenever \ is a factorization partner of \(\mathcal{E}\) (Theorem 2.3). \item[\S 3] gives necessary and sufficient conditions for a compatible equivalence condition \(\backsim\) on \(\mathsf{Span}(\mathcal{C})\) to make its quotient category an allegory (Theorem 3.6). \item[\S 4] shows that the provision \(\mathcal{M}\subseteq\mathsf{Mono}(\mathcal{C})\) is necessary for \(\mathsf{Rel}_{\mathcal{M}}(\mathcal{C})\) to form an allegory (Theorem 4.6). \item[\S 5] shows that, given any stable factorization system \((\mathcal{E},\mathcal{M})\) in a finitely complete category \(\mathcal{C}\), there is a least pullback-stable and composition-closed class \(\mathcal{E}_{\bullet}\) containing \(\mathcal{E}\) and making \(\mathsf{Span}_{\mathcal{E}_{\bullet}}(\mathcal{C})\) a unitary tabular allegory (Theorem 5.8). \item[\S 6] sets up the \(2\)-category of unitary tabular allegories on the one hand and that of finitely complete categories rigged out in a pullback-stable factorization system on the other, showing that the construction of Theorem 5.8 gives rise a left ajoint to the \(2\)-functor \[ \mathsf{Map}:\mathfrak{UTabAll}\rightarrow\mathfrak{STabFact} \] assigning to a unitary tabular allegory its category of Lawverian maps rigged out in its stable factorization system that makes it a regular category. The \textit{Freyd-Scedrov Representation Theorem} [\textit{P. J. Freyd} and \textit{A. Scedrov}, Categories, allegories. Amsterdam etc.: North-Holland (1990; Zbl 0698.18002), 2.154] allows of presenting \(\mathfrak{UTabAll}\) as \(2\)-equivalent to the full subcategory \(\mathfrak{RegCat}\) of \(\mathfrak{STabFact}\) consisting of all regular categories, so that every finitely complete category with a pullback-stable factorization system allows for a reflection into \(\mathfrak{RegCat}\). \end{itemize}