Hadamard Product of Polynomial Functors

The Hadamard Product $P\otimes Q$ of two Polynomial Functors is defined as $\Sigma(p:P(1)).\,\Sigma(q:Q(1)).\,y^{P[p]\times Q[q]}$.

• Note taken on [2025-02-11 Tue 08:33]
  Note that David Spivak calls this the "Dirchlet Product".

The Hadamard product seems to be part of a more general construction that occurs when take the Fibrewise Opposite of a displayed category that has Total Products. This leads to the more general notion of a Hadamard Product in a Cartesian Fibration.

Properties

This induces a Symmetric Monoidal structure on the Category of Polynomial Functors with unit $y$. It is Normal Duoidal with the Composition of Polynomial Functors.