Definition. NP-hardness [np-hard]

A Language $L$ is NP-hard if it is at least as hard as every problem in $NP$, i.e., $\forall A \in NP,A\underset{P}{\leq}L$.

There is nothing in this definition that’s really specific to $NP$. We could switch it out for other complexity classes if we’d like.

Note that we don’t require our language to be in $NP$! $A_{TM}$ is NP-hard.