Full affine equivariance and weak natural transformations in numerical analysis - the case of B-Series

TL;DR

This paper explores the concept of affine equivariance in numerical algorithms, introducing weak natural transformations to better understand their invariance properties, especially in the context of B-Series.

Contribution

It defines weak natural transformations between affine functors and applies this framework to analyze affine equivariance in numerical analysis, particularly for B-Series.

Findings

Identifies limitations of current affine equivariance in algorithms

Introduces the concept of weak natural transformations

Provides examples from numerical analysis and B-Series

Abstract

Many algorithms in numerical analysis are affine equivariant: they are immune to changes of affine coordinates. This is because those algorithms are defined using affine invariant constructions. There is, however, a crucial ingredient missing: most algorithms are in fact defined regardless of the underlying dimension. As a result, they are also invariant with respect to non-invertible affine transformation from spaces of different dimensions. We formulate this property precisely: these algorithms fall short of being natural transformations between affine functors. We give a precise definition of what we call a weak natural transformation between functors, and illustrate the point using examples coming from numerical analysis, in particular B-Series.