Fundamental Group of a Simplicial Space with trivial 0-skeleton

Question

I am trying to find a reference or an explanation for the result that if $X_\bullet$ is a simplicial space with $X_0$ a point then $\pi_1(|X_\bullet|)$ is the free group on $\pi_0(X_1)$ $/ \sim$, where $\sim$ is the relation that $\partial_1(x) = \partial_2(x) \partial_0(x), \forall x\in \pi_0(X_2) $. I think alternatively if I could show that $\pi_1(\sum X_1)$ is the free group on $\pi_0(X_1)$, then this would be sufficient but this doesn't seem to be true as this would imply that the fundamental group of the reduced suspension is always a free group which might not be true (at least to my knowledge). Any suggestions or references would be great.

Answer 1

I don't know about simplicial spaces. For simplicial sets this is first due to Kan, A combinatorial definition of homotopy groups. See section 19. (Wikipedia discusses the case of simplicial complexes, and the ideas should be similar.)