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Many algorithms in numerical analysis are affine equivariant: they are immuneto changes of affine coordinates. This is because those algorithms are definedusing affine invariant constructions. There is, however, a crucial ingredientmissing: most algorithms are in fact defined regardless of the underlyingdimension. As a result, they are also invariant with respect to non-invertibleaffine transformation from spaces of different dimensions. We formulate thisproperty precisely: these algorithms fall short of being naturaltransformations between affine functors. We give a precise definition of whatwe call a weak natural transformation between functors, and illustrate thepoint using examples coming from numerical analysis, in particular B-Series.

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