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Recall that there is an adjunction $$\mathrm{Sd} \dashv \mathrm{Ex} : \mathbf{sSet} \to \mathbf{sSet}$$ where $\mathrm{Sd} (\Delta^n)$ is the first barycentric subdivision of $\Delta^n$. There is a natural transformation $i : \mathrm{id}_{\mathbf{sSet}} \Rightarrow \mathrm{Ex}$, so we get a chain $$X \to \mathrm{Ex} (X) \to \mathrm{Ex}^2 (X) \to \mathrm{Ex}^3 (X) \to \cdots $$ and the simplicial set $\mathrm{Ex}^\infty (X)$ is defined to be the colimit of the above chain.

It is well known that $\mathrm{Ex}^\infty (X)$ is a Kan complex. Let $i^\infty_X : X \to \mathrm{Ex}^\infty (X)$ be the component of the colimiting cocone. Kan [1957, On c.s.s. complexes] (essentially) asserts the following:

Theorem 4.7. If $f : \mathrm{Ex}^\infty (X) \to \mathrm{Ex}^\infty (X)$ is any morphism such that $f \circ i^\infty_X = i^\infty_X$, then $f$ is a weak homotopy equivalence.

Here, ‘weak homotopy equivalence’ means a morphism that induces isomorphisms in $\pi_0$ and all homotopy groups as defined combinatorially for Kan complexes. The proof in op. cit. only shows that we have isomorphisms in $\pi_0$ and $\pi_1$, saying that the case of general $\pi_n$ is "similar although more complicated". How does one carry out this suggestion?

(I am aware of other proofs, but they all involve much heavier machinery. The earliest I know of is [Fritsch and Piccinini, 1990] – 38 years later! – and uses geometric realisation, universal covers, Whitehead's theorem etc.; whereas [Cisinski, 2002, 2006] first establishes the existence of a model structure by very general methods. There really should be a purely combinatorial proof following the pattern of Kan.)

Remark. It suffices to prove the claim in the special case where $f : \mathrm{Ex}^\infty (X) \to \mathrm{Ex}^\infty (X)$ is the morphism $\mathrm{Ex}^\infty (i_X) : \mathrm{Ex}^\infty (X) \to \mathrm{Ex}^\infty (\mathrm{Ex} (X))$, provided we identify $\mathrm{Ex}^\infty (\mathrm{Ex} (X))$ with $\mathrm{Ex}^\infty (X)$ via a suitable isomorphism. In particular, we may assume $f$ is a monomorphism and behaves nicely with respect to the filtration of $\mathrm{Ex}^\infty (X)$ by the canonical embeddings $\mathrm{Ex}^m (X) \to \mathrm{Ex}^\infty (X)$.

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  • $\begingroup$ Could you provide a sketch of how Kan shows that there's an isomorphism in $\pi_0$ and $\pi_1$? $\endgroup$ – user122283 Dec 2 '14 at 17:52

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