Hermitian Categories, Extension of Scalars and Systems of Sesquilinear Forms

Abstract: We prove that the category of systems of sesquilinear forms over a given hermitian category is equivalent to the category of unimodular 1-hermitian forms over another hermitian category. The sesquilinear forms are not required to be unimodular or defined on a reflexive object (i.e. the standard map from the object to its double dual is not assumed to be bijective), and the forms in the system can be defined with respect to different hermitian structures on the given category. This extends a result obtained by E. Bayer-Fluckiger and D. Moldovan. We use the equivalence to define a Witt ring of sesquilinear forms over a hermitian category, and also to generalize various results (e.g.: Witt's Cancelation Theorem, Springer's Theorem, the weak Hasse principle, finiteness of genus) to systems of sesquilinear forms over hermitian categories.