Polynomials as spans

Abstract

The paper defines polynomials in a bicategory $\mathcal{M}$. Polynomials in bicategories $\mathrm{Spn}\mathcal{C}$ of spans in a finitely complete category $\mathcal{C}$ agree with polynomials in $\mathcal{C}$ as defined by Nicola Gambino and Joachim Kock, and by Mark Weber. When $\mathcal{M}$ is \textit{calibrated}, we obtain another bicategory $\mathrm{Poly}\mathcal{M}$. We see that polynomials in $\mathcal{M}$ have representations as pseudofunctors ${\mathcal{M}}^{\mathrm{op}}\to \mathrm{Cat}$. 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".

An anonymous referee wisely required more detail in Examples 12 and 13. This new material is the main reason for the paper expanding to 33 pages. There are some points in that material that I believe are not so well known. The reference list is also expanded

References in corpus (2)

• Polynomial functors and polynomial monads
• Universal properties of bicategories of polynomials

Cited by in corpus (2)

• A diagrammatic approach to symmetric lenses
• Span composition using fake pullbacks