# Full Affine Equivariance and Weak Natural Transformations in Numerical Analysis—The Case of B-Series

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 267)

## 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

Affine Allegory Equivariance Natural transformation

## MSC codes

18B10 58J70 65L06

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