Understanding two proofs in Dwyer and Kan article "Simplicial Localizations"

Question

I can't understand the proofs of propositions 2.6 and 4.2 in https://www3.nd.edu/~wgd/Dvi/SimplicialLocalizations.pdf

We have a category $C$ and a family of maps $W$, and we define the standard resolution $F_*C$ of $C$ as in definition 2.5 and the simplicial localization $LC$ of $C$ with respect to $W$ as in definition 4.1. There is then a natural map $\varphi_*:F_*C \rightarrow C$ defined in 2.5.

Proposition 2.6 states "the map $\varphi_*:F_*C \rightarrow C$ is a weak homotopy equivalence". Proposition 4.2 states "$\pi_0LC=C[W^{-1}] $".

In 2.6 the proof they suggest involves a "contracting homotopy" which I can't define. How can I write this explicitly? And how is it related to the weak equivalence of categories?

In 4.2 they give no indication and I tried to extend the equivalence in 2.6 to an equivalence between the simplicial localization and the ordinary localization, in order to obtain the result simply applying $\pi_0$, but I don't know how to write it down formally.

Answer 1

It's not so bad to prove 4.2 directly in terms of generators and relations. Dwyer and Kan define $LC = F_\ast C [F_\ast W^{-1}]$ where $F_n$ is the $(n+1)$st iteration of the free category comonad. So $\pi_0 LC (X,Y) = F_0 C [ F_0 W^{-1}] / F_1 C [ F_1 W^{-1}]$. Now, $F_0 C[F_0 W^{-1}]$ is the set of paths in $C$ which can go backward along maps in $W$, modulo the equivalence relation identifying $w \bar w = 1$ and $\bar w w = 1$ where $w \in W$ and $\bar w$ denotes $w$ going backwards. There is an obvious map from here to $C[W^{-1}](X,Y)$, so we just have to check that the equivalence relation imposed by the maps from $F_1 C[F_1 W^{-1}]$ is the same as the equivalence relation defining $C[W^{-1}](X,Y)$. The elements of $F_1 C[F_1 W^{-1}](X,Y)$ are like those of $F_0 C[F_0 W^{-1}](X,Y)$, but with an extra layer of parentheses thrown in -- the two maps to $F_0 C[F_0 W^{-1}](X,Y)$ are the map which removes the parentheses, and the map which takes composites within parentheses. These simply impose the relations saying that a path in $C$ is equivalent to its composite, and a backwards path in $W$ is equivalent to its composite, along with concatenations of these rules with each other. These are precisely the relations that hold in $C[W^{-1}](X,Y)$ (in addition to the identifications already made in $F_0 C[F_0 W^{-1}](X,Y)$), so the map $\pi_0 LC(X,Y) \to C[W^{-1}](X,Y)$ is indeed a bijection.