Polynomials as spans

The paper defines polynomials in a bicategory $\mathcal{M}$. Polynomials in bicategories \mathrm{Spn}\mathscr{C} \ of spans in a finitely complete category \mathscr{C} \ agree with polynomials in \mathscr{C} \ as defined by Nicola Gambino and Joachim Kock, and by Mark Weber. When $\mathcal{M}$ is $\mathit{\text{calibrated}}$, we obtain another bicategory $\mathrm{P}\mathrm{o}\mathrm{l}\mathrm{y}\mathcal{M}$. We see that polynomials in $\mathcal{M}$ have representations as pseudofunctors ${\mathcal{M}}^{\mathrm{o}\mathrm{p}}\to \mathrm{C}\mathrm{a}\mathrm{t}$. Calibrations are produced for the bicategory of relations in a regular category and for the bicategory of two-sided modules (distributors) between categories thereby providing new examples of bicategories of “polynomials”.