Abstract

Categorical models of differentiation are either additive categories, that is categories enriched over commutative monoids, or left-additive categories. This is due to the fact that the differentiation of a function depending on a tuple of arguments requires a summation of partial derivatives. From a computational point of view this means that a model of a programming language where programs can be differentiated (in the sense of the differential lambda-calculus) features a strong form of non-determinism. The recently introduced coherent differentiation shows that this is not a fatality, proposing a new setting where morphisms can be differentiated – in a completely standard way – in categories which are not necessarily additive. As an outcome we introduce a differential version of the functional language PCF equipped with a fully deterministic operational semantics. This new approach is based on a functorial axiomatization of the concept of summability.