TITLE: Simplicial replacements in smoothing theory
QUESTION [10 upvotes]: As far as I can tell, ever since Milnor's Microbundles and differentiable structures (1961) paper, whenever people talk about $Diff(\mathbb R^n)$ or $PL(\mathbb R^n)$ or $Homeo(\mathbb R^n)$, they always bring in an auxiliary simplicial space, given various names like the associated "c.s.s. complex" (Milnor), or the associated singular complex $Sing_\cdot(Diff(\mathbb R^n))$.  It seems to me like there are as many names for essentially the same thing as there are authors in this subject. 
My question is, why do people bother with this complex?  i.e. why not just use the original space $Aut^{CAT}(\mathbb R^n)$? 
In Milnor's paper he does not address this question.  It seems to be a trend in the literature to avoid this discussion.  
So to be more precise, I suppose I want to ask the question,

Q: Is there any point in the smoothing literature where the associated simplicial complex to $Aut^{CAT}(\mathbb R^n)$ is absolutely necessary to a proof, and if so, why? 

If there is an affirmative answer to that question, I would like to know what it would take to remove the usage of the associated simplicial complex.  What information are we missing about $Aut^{CAT}(\mathbb R^n)$ that we can avoid when we use the associated simplicial complex? 
The only reference I've found that makes any attempt to address something like this question is Lurie's notes on smoothing theory (lecture 6, bottom of page 1).  But this isn't quite what I'm looking for.

REPLY [3 votes]: I present here a reference for Peter May's comment to Tom Goodwillie's answer in this thread. It also corroborates the comment by John Klein below the question stating that there is no obvious topology on the set of PL homeomorphisms which recovers the correct homotopy type.
Corollary 2 in the article
Ross Geoghegan and William Haver, On the space of piecewise linear homeomorphisms of a manifold, Proceedings of the A.M.S., volume 55, number 1, 1976, pages 145-151
contains the following result:

Let $M$ be a compact PL manifold (without boundary) of dimension different from 4. Let $PLH(M)$ denote the space of PL homeomorphisms of $M$ with the compact-open topology. Further, let $H^\ast(M)$ denote the space of homeomorphisms of $M$ which are isotopic to PL homeomorphisms, equipped with the compact-open topology. In other words, $H^\ast(M)$ is the union of all the path components of the space of homeomorphisms of $M$ which contain a PL homeomorphism. Then the inclusion of $PLH(M)$ into $H^\ast(M)$ is a weak equivalence.

Unfortunately, I am not in a position to say anything about the proof of the above result. Perhaps someone else can provide more information.