Manifold tensor categories

We introduce Manifold tensor categories, which make precise the notion of a tensor category with a manifold of simple objects. A basic example is the category of vector spaces graded by a Lie group. Unlike classic tensor category theory, our setup keeps track of the smooth (and topological) structure of the manifold of simple objects. We set down the necessary definitions for Manifold tensor categories and a generalisation we term Orbifold tensor categories. We also construct a number of examples, most notably two families of examples we call Interpolated Tambara-Yamagami categories and Interpolated quantum group categories. Finally, we show conditions under which pointwise duality data in an Orbifold tensor category automatically assembles into smooth duality data. Our proof uses the classic Implicit function theorem from differential geometry.