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A tangent category alternative to the Faa di Bruno construction

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Jean-Simon P. Lemay

The Faa di Bruno construction, introduced by Cockett and Seely, constructs
a comonad Faa whose coalgebras are precisely Cartesian differential
categories. In other words, for a Cartesian left additive category X,
Faa(X) is the cofree Cartesian differential category over X. Composition
in these cofree Cartesian differential categories is based on the Faa di
Bruno formula, and corresponds to composition of differential forms. This
composition, however, is somewhat complex and difficult to work with. In
this paper we provide an alternative construction of cofree Cartesian
differential categories inspired by tangent categories. In particular,
composition defined here is based on the fact that the chain rule for
Cartesian differential categories can be expressed using the tangent
functor, which simplifies the formulation of the higher order chain rule.

Keywords:
Cartesian Differential Categories, Cofree Cartesian Differential
Categories, Tangent Categories, Higher-Order Chain Rule

2010 MSC:
18A40,18C15,18D99

*Theory and Applications of Categories,*
Vol. 33, 2018,
No. 35, pp 1072-1110.

Published 2018-11-02.

http://www.tac.mta.ca/tac/volumes/33/35/33-35.pdf

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