After-treatment devices are anything in the exhaust that lowers emissions. In general this means SCR (which uses AdBlue) and DPF, but there are also some hybrids like DOC (diesel oxidation catalyst) or oxi-cat. EGR is sometimes thrown into the mix as an outside-the-engine emissions control system, although it doesn't usually need anything special from the oil other than deposit protection.
Of these, the DPF is the most affected by oil type and requires the low SAPS formulations to prolong filter life. SCRs and DOCs can be poisoned by the wrong additives but this is a lesser problem.