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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.

Piotr Pstrągowski
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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 '14 at 19:09
  • By the way, the cellular chain complex of $|Sing(X)|$ coincides with what is usually called the "normalized singular chain complex" of $X$. This is like the usual singular chain complex, except that you're quotienting out by the images of degeneracies. I.e., when passing from the free simplicial abelian group on the singular set to chain complexes, you use the normalized Moore functor instead of the alternating face map functor. – Bruno Stonek Feb 19 '16 at 11:34

1 Answers1

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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.

Tyler Lawson
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