Let $X$ be a based space. Then the Moore loop space $MX$ is defined to be the topological monoid whose points are based loops $[0,a] \to X$ where $a \ge 0$ is allowed to vary. Composition is gotten by concatenating loops.


Since $MX$ is a topological monoid, we can form the *bar construction* $BMX$.
This is geometric realization of the simplicial space
$$
[n] \mapsto (MX)^{\times n}
$$ 
where the face and degeneracy maps are defined using: projection, multiplication, and
insertion of the identity element.


If $X$ is a connected based CW complex, then there is a homotopy equivalence
$$
BMX \simeq X .
$$
A standard way to prove this is to construct a quasi-fibration $EMX \to BMX$ whose
fiber at the basepoint is identified with $MX$ up to homotopy equivalence, 
in which $EMX$ is contractible. The quasi-fibration
is functorially associated with $MX$. However, it seems to me that the equivalence
$BMX \simeq X$ gotten in this fashion depends on a contractible space of choices. In particular it doesn't seem to be natural.


**Question:** 


Is there a zig-zag of *natural transformations* 
$$
BMX = f\_0(X) \leftarrow f\_1(X) \to f\_2(X) \leftarrow \cdots \to f\_n(X) = X
$$
which yields a chain of equivalences when $X$ is a connected CW complex?



On the naturality of the bar construction
I wrote down two quick and simple solutions in Lemmas 14.3 and 15.4 on pages 84 and 90 of 
"Classifying spaces and fibrations ([15] on my web page). The first is the evident zigzag
of natural weak equivalences 
$$ X \leftarrow B(PX,MX,\ast) \rightarrow B(\ast,MX,\ast) = BMX, $$
where $PX$ is the Moore path space.
The left arrow is obtained from an obvious map of simplicial spaces to the constant simplicial
space at $X$: Just compose loops and paths and evaluate at the end point. The second is 
induced from $PX\to \ast$.


The second starts with a very slight modification of what Charles wrote down. For a point 
$u=(t\_0,\cdots, t\_p)\in \Delta^p$ with 
$0\leq t\_i\leq 1$ and $\sum t\_i = 1$, let $u\_{i} = t\_0 + \cdots + t\_{i-1}$. For $\gamma\_i\in MX$
of length $a\_i$, $1\leq i\leq p$, define
$$ \xi |(\gamma\_1,\cdots,\gamma\_p),u| = (\gamma\_1\cdots\gamma\_p)(\sum\_{1\leq i\leq p} u\_i a\_i). $$
This modification makes the compatibility check with the simplicial identities a bit more trivially trivial. To see that $\xi$ is a weak equivalence, take its ordinary loops. There is a natural inclusion $\Omega X \to MX$, which is a weak equivalence; as for
any grouplike monoid, there is a natural weak equivalence $MX \to \Omega BMX$; and we have 
$\Omega \xi\colon \Omega BMX\to \Omega X$. The composite of these three maps is the identity 
map $\Omega X\to \Omega X$, hence $\Omega \xi$ is a weak equivalence. If $X$ is connected, it
follows that $\xi$ is a weak equivalence.


