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Relations for groups and for exact categories.- Galois objects and extensions of Hopf Algebras.- Three dimensional non-abelian cohomology for groups.- H-spaces.- Construction de structures libres.- Categories of group extensions.- Algebras graded by a group.- Diagonal arguments and cartesian closed categories.- Foundations for categories and sets.- On the dimension of objects and categories III Hochschild dimension.- Locally Noetherian categories and generalized strictly linearly compact rings. Applications.- Kan extensions, cotriples and andr¿co) homology.