Categories of modules with differentials (Q1814972)

Before the beginning of the theory of Lie algebroids (1967) and later, many algebraic constructions being substantially some generalizations of the Atiyah sequence of a Lie algebroid have appeared, as, for example, Lie pseudoalgebras (Herz), Lie-Cartan pairs (Kastler, Store), Lie-\(d\)-ring (Palais), \((K\)-\(R)\)-Lie algebras (Rinehart), \((K\)-\(A)\)-linear structures (Bkouche), préespace d'Élie Cartan régulier (Barros) and others (however, the author does not mention all of them). The author makes some ordering in this material by introducing his own general definitions of the category of modules with differentials and the category of vector bundles with differentials containing as subcategories those obtained earlier by other authors. Basing himself on the work by \textit{P. J. Higgins} and \textit{K. Mackenzie} [``Algebraic construction in the category of Lie algebroids'', J. Algebra 129, No. 1, 194-230 (1990; Zbl 0696.22007)] -- being an abstract algebraic treatment of Lie algebroids -- on the ground of these own categories, the author studies the fundamental notions as, for example, morphisms, direct products, pullbacks and others.