|
|
Six-sentence introduction:A bicategory is a kind of (weak) 2-category, and classical Morita theory can be phrased in terms of a classical example of a bicategory. This is the bicategory whose 0-cells are rings, 1-cells are bimodules, and 2-cells are bimodule homomorphisms. The unit 1-cell over a ring is that ring considered as a bimodule over itself, and composition of 1-cells is defined by the tensor product. A Morita equivalence of rings is simply an equivalence of 0-cells in this bicategory. That is, a 1-cell from one ring to another (a bimodule), with a 1-cell inverse, so that the composite (tensor product) of these two bimodules over one ring is isomorphic to the other, and vice-versa. Currently, I'm working to use this bicategorical perspective to give a conceptual unification of various extensions of Morita theory. Notes from my recent talk at the Graduate Student Topology Conference will be available soon. Please contact me with questions or comments. |