Homotopy groups of mapping spaces in a stable $\infty$-category

Question

In the beginning of Higher Algebra, Lurie repeatedly applies whitehead’s theorem for maps between mapping spaces, together with the identity $\pi_n Map_{\mathcal C}(X,Y)=Ext^{-n}(X,Y):=Hom_{h\mathcal C}(X[n],Y)$, where $X[n]=\Sigma^nX$ and these homotopy groups are pointed by the zero map. He uses isomorphisms of all negative Exts to conclude equivalences of mapping spaces. Whitehead’s theorem of course requires a bijection on $\pi_n$ for every $n\geq0$ but also for every base point. So a priori it seems like we haven’t verified the hypotheses of Whitehead’s theorem because we checked for only one base point. Why is this enough? I assume something about the stability of $\mathcal C$ makes the homotopy groups invariant under change of base point, but I don’t see how.

Answer 1

A mapping space in a stable $\infty$-category is a loop space (since $Map(X,Y)\simeq \Omega Map(\Sigma^{-1}X,Y)$), and so in particular it is a group up to homotopy. In particular, all the different components of such a mapping space $M$ are homotopy equivalent: if $M_0$ is the component of the basepoint and $x\in M$ is in a different component, translation by $x$ is a homotopy equivalence from $M_0$ to the component of $x$ (with inverse given by translation by $x^{-1}$). Similarly, any map between these mapping spaces induced by a morphism in the stable $\infty$-category will preserve the group structure up to homotopy, and so will induce the same map (up to homotopy) on the different connected components when we identify them together via translation like this.