Let $\mathcal C$ be a category with pushouts, pullbacks, and a stable factorisation system $(\mathcal E,\mathcal M)$. Let $\mathcal A$ be a subcategory containing all isomorphisms and stable under pushout. Given a span $X \leftarrow N \rightarrow Y$ in $\mathcal A$, we can take the pushout of this span, and then the pullback, to arrive at a canonical map from $N$ to the pullback object $P$:

Pullback of pushout

Under what conditions does this map $N \rightarrow P$ also lie in $\mathcal E$? Is it simpler if we assume $\mathcal A$ contains $\mathcal E$? What about if $\mathcal A = \mathcal E$?

The context is the following. In this paper (behind paywall, sorry), Meisen calls the span $X \leftarrow P \rightarrow Y$ the 'pullback span' of $X \leftarrow N \rightarrow Y$, and shows that an instance of the condition above implies the category with pullback spans as morphisms is equivalent to the category of relations with respect to the factorisation system. You can use this, for example, with epi-mono factorisations in Set to get the category with relations as morphisms.

I'm looking for more examples of categories, like the category of relations, that arise in this way.


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