nLab generalized (Eilenberg-Steenrod) homotopy

The ordinary homotopy groups of a space $X$ are

$$\pi_n(X) = \pi^{S^0}_n(X) = [\Sigma^n S^0, X] = [S^n, X],$$

where $S^0$ is the 0-sphere. We can choose another based space, say $A$. Thus,

$$\pi^{A}_n(X) = [\Sigma^n A, X],$$

are the generalized homotopy groups of $X$ with (co)-coefficients in $A$.

But should this page, mentioning Eilenberg-Steenrod, be about generalized stable homotopy? I.e., should we focus on $\Sigma^n A$ as a spectrum? Mind you, in spectrum it requires $E_n \cong \Omega E_{n + 1}$, where $\Omega$ denotes the based loop space. Don’t we want the requirement $E_{n + 1} \cong \Sigma E_n$? Need to check whether adjunction means this makes no difference.

Tim: To my mind, there should be a spectrum based generalised stable cohomotopy of $X$ 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 $[\Sigma^n A,X]$ but also would ask about not using pointed spaces. The free case is possibly more fun and useful.

Reference

• Hans Baues, Algebraic Homotopy, Cambridge University Press, 1989, p. 117