Definition
A105: invariance is F(g·s) = F(s); equivariance is F(g·s) = ρ_g F(s), where ρ_g acts on the representation. “equivariance may be the more general principle for translation, with invariance as a special case.” Manuscript §4: “For translation, equivariance is often more apt than literal invariance: a verbal joke can become a visual relation conveyed by expressions and gestures. One does not expect equality of coordinates, only corresponding structure.”
What it is not
Not a group action in general: A105 and the manuscript both say translation “is not necessarily an invertible group action”, so the general setting is a semigroup of stochastic maps (CLAIM-098). Equivariance under a semigroup is the manuscript’s “representation-changing” case (ii).