# A theorem of Kan regarding fibrant replacement asked Mar 31, 2014

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)\$.