Bivariations and tensor products

The ordinary tensor product of modules is defined using bilinear maps (bimorphisms), that are linear in each component. keeping this in mind, Linton and Banaschewski with Nelson defined and studied the tensor product in an equational category and in a general (concrete) category K, respectively, using bimorphisms, that is, defined via the Hom-functor on K. Also, the so-called sesquilinear, or one and a half linear maps and the corresponding tensor products generalize these notions for modules and vector spaces. In this paper, taking a concrete category K and an arbitrary subfunctor H of the functor U¢ = Hom ? (Uop ´U) rather than just the Hom-functor, where U is the underlying set functor on K, we generalize sesquilinearity to bivariation and study the related notions such as functional internal lifts, universal bivariants, tensor products, and their interdependence.