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The Faà di Bruno construction

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

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.

Published 2011-09-19.

http://www.tac.mta.ca/tac/volumes/25/15/25-15.dvi

http://www.tac.mta.ca/tac/volumes/25/15/25-15.ps

http://www.tac.mta.ca/tac/volumes/25/15/25-15.pdf

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