  Generalizing Atiyah extensions, we introduce and study differential abelian
tensor categories over differential rings. By a differential ring, we mean a
commutative ring with an action of a Lie ring by derivations. In particular,
these derivations act on a differential category. A differential Tannakian
theory is developed. The main application is to the Galois theory of linear
differential equations with parameters. Namely, we show the existence of a
parameterized Picard-Vessiot extension and, therefore, the Galois
correspondence for many differential fields with, possibly, non-differentially
closed fields of constants, that is, fields of functions of parameters. Other
applications include a substantially simplified test for a system of linear
differential equations with parameters to be isomonodromic, which will appear
in a separate paper. This application is based on differential categories
developed in the present paper, and not just differential algebraic groups and
their representations.
