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We refer to the discussion on different characterizations of the $A_\infty$class of weights, initiated by Duoandikoetxea, Mart\'in-Reyes, and Ombrosi.Twelve definitions of the $A_\infty$ condition are considered. For cubes in$\mathbb{R}^d$ every two conditions are known to be equivalent, while forgeneral bases we have a trichotomy: equivalence, one-way implication, or nodependency may occur. In most cases the relations between different conditionshave already been established. Here all the unsolved cases are treated and, asa result, a full diagram of the said relations is presented.

$A_\infty$ condition for general bases revisited: complete classification of definitions