Suppose $\Sigma$ is an abstract simplicial complex, suppose the vertices are ordered, and let $\tilde{\Sigma}$ be the corresponding simplicial set, where the set of $k$-simplices of $\tilde{\Sigma}$ is the set of $k$-simplices of $\Sigma$, with the obvious face and degeneracy operators. How are the notions of contiguity (as in Spanier, Algebraic Topology, Chapter 3, Section 5) and homotopy in $\mathbf{sSet}$ (as in Goerss and Jardine, Simplicial Homotopy Theory, p. 23) related?
Specifically, consider the abstract simplicial complex on three points $0,1,2$ given by $\Sigma = \{\{0\},\{1\},\{2\}, \{n,n+1\}\}$, for $n \in \{0,1\}$, i.e. the linear graph on three points. The definition in Spanier gives that the identity is contiguous to the constant map $c(\sigma) = \{0\}, \sigma \in \Sigma$. However, I don't see how the definition in Goerss and Jardine gives the corresponding conclusion. Nevertheless, if one considers the simplicial set given by $ST(\tilde{\Sigma})$, where $T$ is the geometric realization functor and $S$ is the singular functor, then I see how one can construct a homotopy between the identity on $ST(\tilde{\Sigma})$ and the constant function, but it seems odd that this wouldn't work directly in $\tilde{\Sigma}$. Is it really the case that $\tilde{\Sigma}$ is not homotopy equivalent to a point in $\mathbf{sSet}$?
Similarly, how does one see that the abstract simplicial complex given by $\mathbb{Z} \cup_{n\in \mathbb{Z}} \{n,n+1\}$ (or the corresponding simplicial set) - whose geometric realization is $\mathbb{R}$ - is homotopy equivalent to a point, as the definition in Spanier only allows for a finite sequence of simplicial maps in the definition of contiguity, and Goerss and Jardine only allow for a single simplicial map in the definition of simplicial homotopy.