The ordinary homotopy groups of a space are
are the generalized homotopy groups of with (co)-coefficients in .
But should this page, mentioning Eilenberg-Steenrod, be about generalized stable homotopy? I.e., should we focus on as a spectrum? Mind you, in spectrum it requires , where denotes the based loop space. Don’t we want the requirement ? Need to check whether adjunction means this makes no difference.
Tim: To my mind, there should be a spectrum based generalised stable cohomotopy of as well perhaps, but the paradigm we have been using has been that it is the spaces that are the first importance here so I would stick with homotopy as but also would ask about not using pointed spaces. The free case is possibly more fun and useful.