Remark 3.2.4.4. In the formulation of Theorem 3.2.4.3, we can replace $(2)$ by the following a priori weaker condition:
- $(2')$
The Kan complex $X$ is connected and, for some choice of base point $x \in X$, the homotopy groups $\pi _{n}(X,x)$ vanish for each $n > 0$.
See Example 3.2.2.18.