The higher order chain rule (categorically)

The higher order chain rule (categorically)

[Robert Seely]

We'd like to announce our paper "The Faa di Bruno construction",
by J.R.B. Cockett and R.A.G. Seely,
a preprint copy of which may be found here:

     http://www.math.mcgill.ca/rags/difftl/faa.pdf

Abstract:

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 Faa 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.

Reference
[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.

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