Full affine equivariance and weak natural transformations in numerical analysis - the case of 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.
Keywords
Cite
@article{arxiv.1605.07344,
title = {Full affine equivariance and weak natural transformations in numerical analysis - the case of B-Series},
author = {Olivier Verdier},
journal= {arXiv preprint arXiv:1605.07344},
year = {2016}
}