If I have an $\infty$-category $\mathcal{C}$ (AKA quasi-category), can I say anything about the homotopy groups of the mapping spaces $\mathrm{Hom}_\mathcal{C}(X,Y)$ for two objects $X$ and $Y$? (These are Kan complexes I believe.) I am trying to develop some intuition for mapping spaces.
All I know is that $\pi_0 \mathrm{Hom}(X,Y)$ is in bijection with morphisms in the homotopy category. What can I say about $\pi_1$, for example?
When $\mathcal C$ is stable, $\pi_n Map_{\mathcal C}(X,Y)$ coincides with $Hom_{h\mathcal C}(X[n],Y)$, where $X[n]$ denotes $\Sigma^nX$, the $n$th power of the suspension functor applied to $X$.