You're talking about Adaboost; general boosting can use any models and the only idea there is that it adds new model to fix residuals of the current chain. BTW "increasingly abstract representation" is a perfect example of this meaningless PR ANNs are built of.
Deep learning is not about fixing the residuals of the current chain. Deep learning isn't even about residuals in the first place. It's about (1) finding good representations of your data (aka feature learning), (2) then adding a discriminative model on top and then (3) tuning everything. There is no relation to boosting at all.
