The Faà di Bruno construction

by J.R.B. Cockett and R.A.G. Seely


In the context of Cartesian differential categories [BCS 09], the structure of the first-order chain rule gives rise to a fibration, the "bundle category". In the present paper we generalise this to the higher-order chain rule (originally developed in the traditional setting by Faà di Bruno in the nineteenth century); given any Cartesian differential category X, there is a "higher-order chain rule fibration" Faà(X) → X over it. In fact, Faà is a comonad (over the category of Cartesian left (semi-)additive categories). Our main theorem is that the coalgebras for this comonad are precisely the Cartesian differential categories. In a sense, this result affirms the "correctness" of the notion of Cartesian differential categories.


[BCS 09] R.F. Blute, J.R.B. Cockett, R.A.G. Seely. "Cartesian differential categories". Theory and Applications of Categories 22 (2009), 622-672.