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Let $X$ be a simplicial set. Recall that we can associate to it a certain topological space, the geometric realization, given by

$ | X | = \int ^{n \in \Delta} X_{n} \times \Delta _{n} \simeq (\bigsqcup X_{n} \times \Delta _{n}) / \sim$,

where on the right hand side we quotient out by both the degeneracy and face maps. In fact, this functor is a part of Quillen equivalence between spaces and simplicial sets.

There is a related construction, that of the fat geometric realization (where I hope I got the terminology right), given by

$ || X || = \int ^{n \in \Delta _{+}} X_{n} \times \Delta _{n} \simeq (\bigsqcup X_{n} \times \Delta _{n}) / \sim$,

where $\Delta _{+} \subseteq \Delta$ is the category of non-empty finite ordinals and injective, order-preserving maps. On the right hand side this amounts to saying that we only quotient out by the face maps. Hence, there is a natural map

$|| X || \rightarrow |X|$.

I am looking for a reference for the fact that this map is a weak equivalence.

I would be ideally interested in a direct argument that does not use the homotopy theory of simplicial sets, but maybe that's asking too much.

I got interested in this statement while thinking of the different ways of proving that weak equivalences of spaces induce isomorphisms in singular homology (this is well-known and there is a direct argument in the textbook of Hatcher). This is obvious for cellular homology, where cellular homology of a general topological space would be cellular homology of some CW-approximation. It seems to me that cellular complex of $| Sing (X) |$ for some space $X$ is very close, but not quite the singular complex of $X$. On the other hand, I believe the cellular complex of $|| Sing (X) ||$ might be exactly the right thing.

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One can just check that this map induces isomorphism on ($\pi_1$ and) homology with arbitrary coefficients (i.e. that the complex generated by degenerate simplices is acyclic) — this is well-known e.g. in the context of Dold-Kan correspondence, I believe. –  Grigory M Jan 13 at 19:09

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up vote 4 down vote accepted

This is proven in Appendix A of Segal's "Categories and cohomology theories". In particular, it is Proposition A.1(iv). Note that in terms of that appendix, a simplicial set is a particular instance of a simplicial space, and it always satisfies the "goodness" criterion described in A.4 which guarantees that the natural map $||X|| \to |X|$ is an equivalence.

In your situation, $||X||$ and $|X|$ are both CW-complexes, and so it is also a homotopy equivalence.

Note that in A.5 there are some equivalence signs that are hard to see.

You are correct that the cellular chain complex of $||Sing(X)||$ is the standard chain complex computing $H_*(X)$. The cellular chain complex of $|Sing(X)|$ is a direct summand of it, and the complementary summand (the complex of degenerate simplices) always has trivial homology.

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