In homotopy type theory, the Kleisli category of the exponential comonad on spans is equivalent to the category of V-ary polynomials, yielding a new model of linear logic.
Polynomials as spans
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abstract
The paper defines polynomials in a bicategory $\mathscr{M}$. 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 $\mathscr{M}$ is \textit{calibrated}, we obtain another bicategory $\mathrm{Poly}\mathscr{M}$. We see that polynomials in $\mathscr{M}$ have representations as pseudofunctors $\mathscr{M}^{\mathrm{op}}\to \mathrm{Cat}$. 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".
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2024 1verdicts
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Polynomials in homotopy type theory as a Kleisli category
In homotopy type theory, the Kleisli category of the exponential comonad on spans is equivalent to the category of V-ary polynomials, yielding a new model of linear logic.