## Polynomials in Symmetric Monoidal Categories

### Matthew di Meglio – 27 January 2021, 3 February 2021

Weber studied polynomials and polynomial functors in a category C with pullbacks, introducing the concept of a distributivity pullback to help describe polynomial composition. Grunenfelder and Paré explored how J-comodules, where J is a K-coalgebra, may be thought of as J-indexed families of vector spaces over a field K. Using ideas from Aguiar’s PhD thesis, much of Grunenfelder and Paré’s work generalises to the setting of a symmetric monoidal category with equalisers preserved by tensoring. Regarding C as a cartesian monoidal category (assuming additionally that C has a terminal object), the generalised theory of Grunenfelder and Paré for C reduces to the usual theory of C as a C-indexed category. This observation hints that the notion of polynomials and polynomial functors may generalise to the setting of sufficiently nice symmetric monoidal categories.

Part I will introduce all of the key concepts so that we may define generalised polynomials, their associated polynomial functors, and their composition. Part II will focus on my work towards showing that the mapping from polynomials to polynomial functors is functorial, ending with a discussion of some of the needed results which don’t seem to easily generalise.

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