Recall two facts about that geometric realization $T:S\to Top$ :

(1) $T$ preserves all colimits.

(2) $T$ preserves finite limits.

From 1.Thus, we see that it is enough to show that if $f:X\to Y$ is a map of simplicial sets such that $T(f)$ is a bijection, then $f$ is an isomorphism.

(3) If $f:X\to Y$ is a map of simplicial sets such that, for every non-degenerate simplex $t$ of $Y_n$, there is a simplex $s$ of $X_n$ such that $f(s)=t$, then $f$ is surjective.

(4) If

Let $f:X\to Y$ is be a map of simplical sets such that (i) if $s\in X_n$ is non-degenerate, then $f(s)\in Y_n$ is non-degenerate, and (ii) if $s,s'\in X_n$ are non-degenerate and $f(s)=f(s')$, then $s=s'$, then s=s'$. It follows that $f$ is injective.

If $f:X\to Y$ is a map of simplicial sets such that for each non-degenerate $t\in Y_k$, there exists a non-degenerate $s$ in some $X_n$ such that $f(s)$ is a degeneracy of $t$ (or $f(s)=t$, when $n=k$), then $f$ is surjective.

$$ TX \approx \bigcup_n \bigcup_\sigma (\text{interior \Delta^n-\partial\Delta^n),$$where the $\sigma$ range over non-degenerate $n$-simplices. It is not hard to see how this behaves as a functor: a map $f:X\to Y$ induces a map $Tf:TX\to TY$ which sends the boundaryless simplex of corresponding to a non-degenerate $\Delta^n$}).$$\sigma\in X_n$ to the bondaryless simplex corresponding to the non-degenerate $\tau\in Y_k$, where $\tau$ the non-degenerate simplex of which $f(\sigma)$ is degenerate (or, if $f(\sigma)$ is non-degnererate, then $f(\sigma)=\tau$.)The resulting map of boundaryless simplices $(\Delta^n-\partial \Delta^n)\to (\Delta^k-\partial \Delta^k)$ is described by a surjective map $[n]\to [k]$ in $\Delta$, and can be bijective only if $n=k$.

Given this, it's easy to check that if $Tf$ is a bijection, then the conditions of (1) and (2) must be satisfied, so $f$ must be bijective too.

I claim that geometric realization reflects colimits of all sorts. That is, given a functor $F: C\to sSet$ and a cocone ${F(C_i)\to X}$ in $sSet$ such that $\newcommand{\colim}{\operatorname{colim}}T(\colim_C F)\to T(X)$ is a bijection, then $\colim_C F\to X$ is an isomorphism of simplicial sets.

Recall two facts about geometric realization $T:S\to Top$:

• (1) $T$ preserves colimits.
• (2) $T$ preserves finite limits.

From 1., we see that it is enough to show that if $f:X\to Y$ is a map of simplicial sets such that $T(f)$ is a bijection, then $f$ is an isomorphism.

To show this, I want to use the notion of a nongenerate simplex of a simplicial set $X$, and the fact that every simplex $x\in X_n$ is the degeneracy of a unique non-degenerate simplex $t\in X_k$, and is so in a unique way (i.e., there is a unique surjective map $\sigma:[n]\to [k]$ in $\Delta$ such that $(X\sigma)(t)=x$; this fact is sometimes called the "Eilenberg-Zilber lemma".)

Given this, it is not hard to show the following.

• (3) If $f:X\to Y$ is a map of simplicial sets such that, for every non-degenerate simplex $t$ of $Y_n$, there is a simplex $s$ of $X_n$ such that $f(s)=t$, then $f$ is surjective.
• (4) If $f:X\to Y$ is a map of simplical sets such that (i) if $s\in X_n$ is non-degenerate, then $f(s)\in Y_n$ is non-degenerate, and (ii) if $s,s'\in X_n$ are non-degenerate and $f(s)=f(s')$, then $s=s'$, then $f$ is injective.

Now consider the geometric realization $TX$ of a simplicial set $X$. As a set, this has the form of a disjoint union $$ TX \approx \bigcup_n \bigcup_\sigma (\text{interior of $\Delta^n$}).$$