Cartesian differential categories

This paper revisits the authors’ notion of a differential category from a different perspective. A differential category is an additive symmetric monoidal category with a comonad (a “coalgebra modality”) and a differential combinator. The morphisms of a differential category should be thought of as the linear maps; the differentiable or smooth maps would then be morphisms of the coKleisli category. The purpose of the present paper is to directly axiomatize differentiable maps and thus to move the emphasis from the linear notion to structures resembling the coKleisli category. The result is a setting with a more evident and intuitive relationship to the familiar notion of calculus on smooth maps. Indeed a primary example is the category whose objects are Euclidean spaces and whose morphisms are smooth maps. A Cartesian differential category is a Cartesian left additive category which possesses a Cartesian differential operator. The differential operator itself must satisfy a number of equations, which guarantee, in particular, that the differential of any map is “linear” in a suitable sense. We present an analysis of the basic properties of Cartesian differential categories. We show that under modest and natural assumptions, the coKleisli category of a differential category is Cartesian differential. Finally we present a (sound and complete) term calculus for these categories which allows their structure to be analysed using essentially the same language one might use for traditional multi-variable calculus. 0. Introduction Over the past few centuries, one of the most fundamental concepts in all of mathematics has been differentiation. In recent decades several attempts have been made to abstract this notion, including approaches based on geometric, algebraic, and logical intuitions. The approach of the current paper is categorical, insofar as we wish to consider axiomatizations of categories which have sufficient structure to define differentiation of maps. Any additional categorical structure, e.g. monoidal, Cartesian, or comonadic, exists in support of differentiation. In [BCS 06], the current authors introduced the notion of a differential category to provide a basic axiomatization for differential operators in monoidal categories. The initial impetus for the definition came from work of Ehrhard and Regnier [Ehrhard & Regnier 05, Ehrhard & Regnier 03], who defined first a notion of differential λ-calculus and subsequently differential proof nets. The differential λ-calculus is an extension of simply-typed λ-calculus with an additional operation allowing the differentiation of terms. Differential proof nets are a (graph-theoretic) syntax for linear logic extended with a differential operator on proofs. An important feature of their systems, not precluded in ours, is that in one setting, they combine the essence of both calculus and computability. c © R.F. Blute, J.R.B. Cockett and R.A.G. Seely, 2008. Permission to copy for private use granted.