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The category of fat extensions is introduced, as well as the category of abstract $2$-term representations up to homotopy (ruths) -- the intrinsic objects behind usual (split) $2$-term ruths. We obtain a one-to-one correspondence between them, and relate to the well-known equivalence between $2$-term ruths and VB-groupoids/algebroids. On the other hand, we show that fat extensions of groupoids correspond to general linear PB-groupoids. The differentiation procedure of fat extensions is disc"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We obtain a one-to-one correspondence between [the category of fat extensions and the category of abstract 2-term representations up to homotopy], and relate to the well-known equivalence between 2-term ruths and VB-groupoids/algebroids. 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