Structures on Categories of Polynomials

Abstract

We define the monoidal category $(Pol{y}_E,y,◃)$ of polynomials under composition in any category $E$ with finite limits, including both cartesian and vertical morphisms of polynomials, and generalize to this setting the Dirichlet tensor product of polynomials $\otimes$, duoidality of $\otimes$ and $◃$, closure of $\otimes$, and coclosures of $◃$. We also prove that $◃$-comonoids in $Pol{y}_E$ are precisely the internal categories in $E$ whose source morphism is exponentiable, generalizing a result of Ahman-Uustalu equating categories with polynomial comonads, and show that coalgebras in this setting correspond to internal copresheaves. Finally, the double category of ``typed'' polynomials in $E$ is recovered using $◃$-bicomodules in $Pol{y}_E$.

50 pages. Additional references added, and minor expository improvements