Crossed modules and internal categories of Lie-Rinehart algebras

We give a definition of internal crossed module in a protomodular, Barr-exact category C with finite coproducts, and we show that the category of internal crossed modules in C is equivalent to the category of internal categories in C . A category that is protomodular, Barr-exact, has finite coproducts, and is also pointed is, equivalently, a semi-abelian category. Our definition of internal crossed module is a generalisation of a definition of crossed module in a semi-abelian category due to Janelidze. Similarly, our theorem stating the equivalence of the categories of internal crossed modules and internal categories in C is a generalisation of a corresponding theorem of Janelidze’s. We show that the category LR of Lie-Rinehart algebras is protomodular, Barr-exact, and has finite coproducts,showing that our new definition of internal crossed module applies to the category of Lie-Rinehart algebras, and thus that the categories of internal crossed modules in LR and internal categories in LR are equivalent. We then compare our definition of internal crossed module with the existing definition of a crossed module of Lie-Rinehart algebras.