In P.May's book "A concise course in Algebraic Topology", chapter 16, He establishes a weak equivalence between $\Gamma X = |S_*(X)|$ and $X$, where $X$ is a topological space, $S_*(X)$ is the total singular complex, and $||$ denoting the geometric realization.
The following is the proof in the book,
We must still explain why γ : ΓX −→ X is a weak equivalence. In fact, it is tautologically obvious that γ induces an epimorphism on all homotopy groups: a map of pairs\begin{equation} f : (\Delta _n, \partial \Delta_n) \rightarrow (X, x) \end{equation} determines the map of pairs \begin{equation} \tilde{f} : (\Delta _n, \partial \Delta_n) \rightarrow (ΓX, |x, 1|) \end{equation} specified by $\tilde{f}(u) = |f,u|$, and $γ ◦\tilde{f} = f$. Injectivity is more delicate, and we shall only give a sketch. Given a map $ g : (\Delta _n, \partial \Delta_n) \rightarrow (ΓX, |x, 1|)$, we may first apply cellular approximation to obtain a homotopy of $g$ with a cellular map and we may then subdivide the domain and apply a further homotopy so as to obtain a map g′ ≃ g such that g′ is simplicial, in the sense that g′ ◦ e is a cell of ΓX for every cell e of the subdivision of $\Delta_n$. Suppose that γ ◦ g and thus γ ◦ g′ is homotopic to the constant map $c_x$ at the point $x$. We may view a homotopy $h:γ◦g′ ≃c_x$ as a map \begin{equation} h:(\Delta_n ×I,∂\Delta_n ×I∪\Delta_n ×{1})\rightarrow(X,x) \end{equation} We can simplicially subdivide $\Delta_n × I$ so finely that our subdivided $\Delta_n = \Delta_n × {0}$ is a subcomplex. We can then lift $h$ simplex by simplex to a simplicial map \begin{equation} \tilde{h}:(\Delta_n ×I,∂\Delta_n ×I∪\Delta_n ×{1})\rightarrow(ΓX,|x,1|) \end{equation} such that $\tilde{h}$ restricts to $\tilde{g}'$ on $\Delta_n×{0}$ and γ◦h=h.
I can't understand the injectivity part.. Especially the subdivision and lifting.