A characterization of weakly Schreier extensions of monoids

A split extension of monoids with kernel $k \colon N \to G$, cokernel $e \colon G \to H$ and splitting $s \colon H \to G$ is Schreier if there exists a unique set-theoretic map $q \colon G \to N$ such that for all $g \in G$, $g = kq(g) \cdot se(g)$. Schreier extensions have a complete characterization and have been shown to correspond to monoid actions of $H$ on $N$. If the uniqueness requirement of $q$ is relaxed, the resulting split extension is called weakly Schreier. A natural example of these is the Artin glueings of frames. In this paper we provide a complete characterization of the weakly Schreier extensions of $H$ by $N$, proving them to be equivalent to certain quotients of $N \times H$ paired with a function that behaves like an action with respect to the quotient. Furthermore, we demonstrate the failure of the split short lemma in this setting and provide a full characterization of the morphisms that occur between weakly Schreier extensions. Finally, we use the characterization to construct some classes of examples of weakly Schreier extensions.