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Cartesian differential categories come equipped with a differentialcombinator that formalizes the derivative from multi-variable differentialcalculus, and also provide the categorical semantics of the differential$\lambda$-calculus. An important source of examples of Cartesian differentialcategories are the coKleisli categories of the comonads of differentialcategories, where the latter concept provides the categorical semantics ofdifferential linear logic. In this paper, we generalize this construction byintroducing Cartesian differential comonads, which are precisely the comonadswhose coKleisli categories are Cartesian differential categories, and thusallows for a wider variety of examples of Cartesian differential categories. Assuch, we construct new examples of Cartesian differential categories fromCartesian differential comonads based on power series, divided power algebras,and Zinbiel algebras.

Cartesian Differential Comonads and New Models of Cartesian Differential Categories