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
[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.