Split Lemma and First Isomorphism Theorem for groupoids

Groupoids are the oidification of groups, and they are largely used in topology and representation theory. We consider here the category $\mathsf{Gpd}$ of all groupoids with all morphisms, and the category $\mathsf{Gpd}_\Lambda$ of groupoids over a fixed set of vertices $\Lambda$, with morphisms fixing $\Lambda$. In $\mathsf{Gpd}_\Lambda$, a First Isomorphism Theorem is already well known; see \'Avila, Mar\'in, and Pinedo (2020). Famously, the First Isomorphism Theorem fails to hold in $\mathsf{Gpd}$. However, we retrieve here a universally lifted version of the First Isomorphism Theorem in $\mathsf{Gpd}$, through the definition of virtual kernels. Semidirect products of a group by a groupoid are well known. We define crossed products in $\mathsf{Gpd}$, and prove that they are equivalent to split epimorphisms, i.e. that they are the `categorial' notion of semidirect product in $\mathsf{Gpd}$ in the sense of Bourn and Janelidze (1998). We observe that in $\mathsf{Gpd}_\Lambda$ crossed products and semidirect products are essentially equivalent, under mild assumptions, and our Split Lemma in $\mathsf{Gpd}$ collapses to a much simpler Split Lemma in $\mathsf{Gpd}_\Lambda$ that appears in Metere and Montoli (2010) and Ibort and Marmo (2023).