In the context of Cartesian differential categories, 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'' Faa(X) --> X over it. In fact, Faa 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.
Keywords: Higher-order chain rule, Cartesian differential categories, bundle fibration, coalgebras
2000 MSC: 18D10, 18C20, 12H05, 32W99
Theory and Applications of Categories, Vol. 25, 2011, No. 15, pp 393-425.