A theorem of Kan regarding fibrant replacement

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asked Mar 31, 2014 by Zhen Lin

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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