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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.
@article{TAC_2011_25_a14,
author = {J.R.B. Cockett and R.A.G. Seely},
title = {The {Fa\`a} di {Bruno} construction},
journal = {Theory and applications of categories},
pages = {393--425},
publisher = {mathdoc},
volume = {25},
year = {2011},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TAC_2011_25_a14/}
}
J.R.B. Cockett; R.A.G. Seely. The Faà di Bruno construction. Theory and applications of categories, Tome 25 (2011), pp. 393-425. http://geodesic.mathdoc.fr/item/TAC_2011_25_a14/