Semidirect products and crossed modules in monoids with operations
Internal categories in the category of groups correspond to crossed modules (of groups). The same type of equivalence works in categories of groups with operations, but fails in general in weaker algebraic contexts such as that of monoids. \textit{A. Patchkoria} [Georgian Math. J. 5, No. 6, 575--581 (1998; Zbl 0915.18002)] introduced a particular kind of internal category in the category of monoids equivalent to crossed modules of monoids. This paper generalises Patchkoria's results and characterises those internal categories in categories of monoids with operations that correspond to a sensible notion of crossed module in those contexts.
scientific article; zbMATH DE number 6155422