Define a reflexive relation on the set of zero-simplices of a simplicial set $A$ by saying that $x\sim y$ iff there is a one-simplex $h$ with $0$-face $y$ and $1$-face $x$. This is not an equivalence relation in general but it is so if $A$ is Kan. Define $\pi_0(A)=A_0/\sim$ in this case.

Let $x:\Delta^0\to A$ be a zero-simplex, $n\in\mathbb{N}$ and consider the pullback

\begin{array}{rcl} A(n,x) &\to& Map(\Delta^n,A)\\ \downarrow &&\downarrow\\ Map(\partial\Delta^n,\Delta^0)&\to&Map(\partial\Delta^n,A) \end{array}

induced by the obvious maps and set $\pi_n(A,x)=\pi_0(A(n.x))$ as the simplicial homotopy groups. This works since $A(n,x)$ is Kan if $A$ is so.

An important theorem states that $\pi_n(A,x)=\pi_n(|A|,|x|)$ the bars denoting the realization functor adjoint to the singular functor $S$. The homotopy category of the usual model structure on simplicial sets is given by inverting the "weak equivalences", i.e. the maps $f:A\to B$ such that $\pi_n(S(f),x)$ is an isomorphism for all $n$ and all basepoints. One has to apply $S$ here to make things Kan.

Does one get the same homotopy category if one lets $\pi_0$ be the equivalence classes of the equivalence relation **generated** by $\sim$? One can define all necessary concepts exactly as above without demanding $A$ to be Kan. Does one get the same homotopy category then?