Effective homotopy in a Kan fibration

Definition 1.1. A simplicial set K is said to satisfy the extension condition if for every collection of n+ 1 n-simplices x0, x1, . . . , xk−1, xk+1, . . . , xn+1 which satisfy the compatibility condition ∂ixj = ∂j−1xi for all i < j, i 6= k, and j 6= k, there exists an (n+ 1)-simplex x ∈ Kn+1 such that ∂ix = xi for every i 6= k. A simplicial set which satisfies the extension condition is called a Kan simplicial set or a Kan complex.