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.