Theorem 3.2.4.3. Let $X$ be a Kan complex. The following conditions are equivalent:
The Kan complex $X$ is contractible.
The Kan complex $X$ is connected and the homotopy groups $\pi _{n}(X,x)$ vanish for each $n > 0$ and every choice of base point $x \in X$.
The projection map $X \rightarrow \Delta ^0$ is a trivial Kan fibration of simplicial sets.
Proof of Theorem 3.2.4.3. Let $X$ be a Kan complex. We wish to show that the following conditions are equivalent:
The Kan complex $X$ is contractible: that is, the projection map $X \rightarrow \Delta ^0$ is a homotopy equivalence.
The Kan complex $X$ is connected and the homotopy groups $\pi _{n}(X,x)$ vanish for every integer $n > 0$ and every vertex $x \in X$.
The projection map $X \rightarrow \Delta ^0$ is a trivial Kan fibration: that is, every morphism of simplicial sets $\operatorname{\partial \Delta }^{m} \rightarrow X$ can be extended to an $m$-simplex of $X$.
The implication $(1) \Rightarrow (2)$ follows from Remark 3.2.2.17, and the implication $(3) \Rightarrow (1)$ is a special case of Proposition 3.1.6.10. The equivalence of $(2)$ and $(3)$ follows from Lemma 3.2.4.13 and Variant 3.2.4.14. $\square$