There is a natural way to convert a simplicial complex $C$ into an "equivalent" simplicial set $S$: after ordering the vertices, the simplices in $C$ correspond exactly to the non-degenerate simplices of $S$.
If $C$ is finite and connected, it can easily be converted to a $0$-reduced sset $S$ (that is, having only one vertex) by factoring out a maximal tree in the $1$-skeleton: the underlying topological spaces have the same homotopy type.
Can a similar method be applied to convert a simply connected complex $C$ into a $1$-reduced sset (that is, having only one vertex and only one $1$-simplex)? I guess that for this, we need to have an algorithm for contracting loops in $C$. Let's say that for each loop $\gamma$, a simplicial map $f_{\gamma}$ is given from a subdivision $D$ of a $2$-disc into $C$ such that $f|_{\partial D}=\gamma$ (or something like that).
Is there an algorithmic way how to use the loop-contractions to convert $S$ into something $1$-reduced and of the same homotopy type?
The intuition is simple: "factor out the filler of each loop". But I have problem formalizing it. It seems to me that the filler can be complicated, self-intersecting, etc, not anything that simple as a spanning tree.
For a simply connected complex $C$, its underlying space $|C|$ is a topological space.
For any pointed, connected topological space $X$, one can define its Eilenberg subcomplex $$\bar{S}_n(X):=\{f:\Delta^n\to X\mid f(v_i)=* \text{ for all vertices }v_i\in\Delta^n\}$$ which is a simplicial set. They have the same homotopy type.
Thus, what you need to do is to take a basepoint (a fixed vertex) and apply the Eilenberg subcomplex construction.