Let $G$ be a (**Edit:** path-)connected topological group. Under what additional hypotheses on $G$ is it true that $LBG$ is a classifying space for $LG$? (or, I guess equivalently, when is $LBG \sim BLG$?) Here I'm taking the **free** loop space, and the compact-open topology on it. I know it is true for strong hypotheses, such as $G$ a compact Lie group. If one can find a model of $BG$ that is locally contractible and paracompact, then by Atiyah and Bott's *The Yang-Mills Equations over Riemann Surfaces* (doi:[10.1098/rsta.1983.0017](https://doi.org/10.1098/rsta.1983.0017)), Proposition 2.4, I believe it is possible. So, alternatively, for what $G$ is it true that $BG$ can be thus chosen?



For which G is BLG weak homotopy equivalent to LBG?
[UPDATE: There were some mistakes in the first version. Here is a more careful account.]


I'll work everywhere with CGWH spaces, so I have a Cartesian closed category.


Note that $BLG$ is always path-connected, but $\pi\_0(LBG)=\pi\_0(G)/\text{conjugacy}$, so we need to assume that $G$ is path-connected. (The question says connected, but this may be a bit stronger; I do not know whether there are connected topological groups that are not path-connected.) 


For any space $X$ and any $t\in S^1$ we have an evaluation map $\epsilon\_{X,t}\colon LX\to X$. For the special case of the basepoint $1\in S^1$ we write $p\_X=\epsilon\_{X,1}\colon LX\to X$. This is always a Hurewicz fibration. If $X$ is based we have a fibre sequence $\Omega X\xrightarrow{i\_X}LX\xrightarrow{p\_X}X$.


Now let $G$ be a topological group. We write $EG$ and $BG$ for the usual simplicial constructions, so $EG$ is contractible and has a free $G$-action with orbit space $BG$. Simplicial methods also give a natural commutative diagram 
$\require{AMScd}$
\begin{CD}
G @>k\_{G}>> EG @>r\_{G}>> BG\\
@Vj\_GVV @VV l\_G V @VV 1 V\\
\Omega BG @>>> PBG @>>> BG
\end{CD} 
Both $EG$ and $PBG$ are contractible, and the bottom row is a Hurewicz fibration. If the top row is also a Hurewicz fibration, we can conclude that $j\_G\colon G\to\Omega BG$ is a homotopy equivalence. If the top row is merely a Serre fibration or quasifibration, we can still conclude that $j\_G$ is a weak equivalence. I do not know what are the minimal conditions for the top row to be a quasifibration. 


Next, given $u\in BLG$ and $t\in S^1$ we have a homomorphism $\epsilon\_{G,t}\colon LG\to G$ and thus a map $B\epsilon\_{G,t}\colon BLG\to BG$ and thus an element $(B\epsilon\_{G,t})(u)\in BG$. We would like to define $f\_G\colon BLG\to LBG$ by $(f\_G(u))(t)=(B\epsilon\_{G,t})(u)$. To justify this we need to check continuity in $t$ and then in $u$. This in turn needs some continuity properties of the functor $B$, which can be proved using some abstract nonsense with CGWH spaces and Cartesian closure. (The initial version of this answer referred to a natural map in the opposite direction, but I think that does not actually exist.) We now want to construct a diagram as follows:
\begin{CD}
 B\Omega G @>Bi\_G>> BLG @>Bp\_G>> BG \\
 @V f'\_G VV @V f\_G VV @VV 1 V \\
 \Omega BG @>>i\_{BG}> LBG @>>p\_{BG}> BG 
\end{CD}
Constructions that we have already discussed provide all spaces and maps except for $f'\_G$. On the top row we note that $(Bp\_G)\circ (Bi\_G)$ is trivial, and on the bottom row we know that $i\_{BG}$ is the fibre of $p\_{BG}$, so there is a unique way to fill in $f'\_G$. The bottom row is always a Hurewicz fibration. The top row is obtained by applying $B$ to a Hurewicz fibration of topological groups, but it is not clear exactly what we get from that. If the top row is at least a Serre fibration, we see that $f\_G$ is a weak equivalence iff $f'\_G$ is a weak equivalence.


Finally, define $\tau\colon S^1\wedge S^1\to S^1\wedge S^1$ by $\tau(s\wedge t)=t\wedge s$. From the definitions one can check that the following diagram commutes:
\begin{CD}
 \Omega G @>j\_{\Omega G}>> \Omega B\Omega G \\
 @V \Omega j\_G VV @VV \Omega f'\_G V \\
 \Omega^2 BG @>>\tau^\*> \Omega^2 BG 
\end{CD}


If $j\_G$ and $j\_{\Omega G}$ are weak equivalences, we conclude that $f'\_G$ is also a weak equivalence.


All this assumes that we start with the simplicial definition of $BG$. One could instead consider an axiomatic characterisation of $BG$, which might include the condition that the map $EG\to BG$ is a Serre fibration. The idea should be that $[X,BG]$ should biject with the set of isomorphism classes of principal $G$-bundles over $X$, but one would need to restrict attention to paracompact $X$, or to principal bundles over arbitrary $X$ that admit a numerable trivialising cover. I do not know how the technicalities would work out.


