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It is a well-known fact that the category $\mathsf{Cat}(\mathbf{C})$ ofinternal categories in a category $\mathbf{C}$ has a description in terms ofcrossed modules, when $\mathbf{C}=\mathbf{Gr}$ is the category of groups. Theproof of this result heavily uses the fact that any split epimorphismdecomposes as a semi-direct product. An equivalent statement does not hold inthe category $\mathbf{Mon}$ of monoids. In a previous work on quadraticalgebras, I constructed an internal category in the category of monoids, seeSection 6. Based on this construction, this paper will introduce the notion ofa crossed semi-bimodule and show that it gives rise to an object in$\mathsf{Cat}(\mathbf{Mon})$. I will also relate this new notion to the crossedsemi-modules introduced earlier by A. Patchkoria.

On internal categories and crossed objects in the category of monoids