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$\DeclareMathOperator\Sing{Sing}$I'm writing a paper about simplicial sets and how they may “replace” simplicial complexes in some known results. To do this, we need to check that they induces the same topological space when realized.

Let $K$ be a simplicial complex with vertex set $V$, and choose an ordering on $V$. We construct a simplicial set $\Sing(K)$ by letting the $n$-simplices be order-preserving maps $[n]\to V$, such that the image is a simplex of $K$. You can read more about this here.

It is a well known fact that the geometric realizations $|K|$ and $|\Sing(K)|$ are homeomorphic, yet I can’t find any reference to include in my paper.

My question: Do any of you know about an article/book proving this? I have seen proofs, but not in a published paper.

Or, do any of you know about a “short” and neat proof of this?

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    $\begingroup$ What, concretely, is your definition of $\left| K \right|$? (At least for finite $K$, there is a definition as a subspace of $\mathbb{R}^V$, but it feels like it would be a nightmare to prove anything using this definition.) $\endgroup$
    – Zhen Lin
    Commented Nov 2, 2022 at 12:02
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    $\begingroup$ The key point I think is that for the simplicial set you construct the following two things are true. Each face of a nondegenerate simplex is nondegenerate and a nondegenerate simplex is determined by its vertices. From the first property I think you can see the geometric realization is just gluing simplices along faces and the second guarantees the intersection of two simplices is another simplex or something to that effect. $\endgroup$
    – Benjamin Steinberg
    Commented Nov 2, 2022 at 12:35

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