nLab
generalized (Eilenberg-Steenrod) homotopy

The ordinary homotopy groups of a space $X$ are

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

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