Theorem 3.2.6.1. Let $f: (X,x) \rightarrow (S,s)$ be a Kan fibration between pointed Kan complexes. Then the sequence of pointed sets
\[ \cdots \rightarrow \pi _{2}(S,s) \xrightarrow {\partial } \pi _{1}(X_ s, x) \rightarrow \pi _{1}( X, x) \rightarrow \pi _1(S,s) \xrightarrow {\partial } \pi _{0}(X_ s, x) \rightarrow \pi _0( X,x) \rightarrow \pi _0(S,s) \]
is exact; here $\partial : \pi _{n+1}(S,s) \rightarrow \pi _{n}(X_ s, x)$ denotes the connecting homomorphism of Construction 3.2.5.3.