Noncommutative geometry and differential calculus in tensor categories.

Many of the fundamental geometric concepts in the theory of smooth manifolds can be rephrased algebraically in terms of the commutative algebra of smooth functions on the manifold. We will discuss how and why one might try to generalize these ideas to noncommutative algebras via the notion of a differential calculus, and some fundamental difficulties with the standard approaches.  In the case our algebra in question exhibits categorical symmetries, we will show how many of these difficulties can be overcome by applying constructions internal to an abstract tensor category.