We refer to the discussion on different characterizations of A ∞ A_\infty class weights, initiated by Duoandikoetxea, Martín-Reyes, and Ombrosi [Math. Z. 282 (2016), pp. 955–972]. Twelve definitions condition are considered. For cubes in alttext="double-struck upper R Superscript d"> mathvariant="double-struck">R d encoding="application/x-tex">\mathbb {R}^d every two conditions known be equivalent, while for general bases we have a trichotomy: equivalence, one-way implication, or no dependency may occur. In most cases relations between already been established. Here all unsolved treated and, as result, full diagram said is presented.