I have defined polynomial in a calibrated bicategory M; it is a span with one very good leg and the other fairly good. (In reference to Michael Johnson's ACS talk on 12 Feb and the work of Bryce Clarke, fairly good can be no restriction.)
The calibration allows polynomials to compose as spans. Tabulations from the terminal object in a bicategory M give rise to a calibration. The guiding example of this phenomenon was the bicategory of spans in a finitely complete category. I shall give two other examples and reinterpret the bicategories of polynomials in terms of Kleisli bicategories.