I'm having trouble with the following elementary(?) thing, but its confusing me alot.
For a category $\mathcal{C}$, we can define the classifying space in terms of nerve.
Similarly, for a topological group (or monoid) G, we can think of it as a category with one point, say $*$.
So, $B_{n}G$ is the set of all $* \xrightarrow{g_0} * \xrightarrow{g_1} \cdots \xrightarrow{g_n} *$ (where $g_i \in G$) or as a tuple $(g_0, g_1, \cdots ,g_n)$, i.e. $B_{n}G = G^{n+1}$.
To get its classifying space $BG$, we can take its geometric realization.
The question that I'm bothered with is, how can we think (or describe) the loops in $BG$, that is, how can we describe the elements of $\Omega BG$, explicitly?
I think we could think of it as tuples $(g_0, ... g_n)$ such that $g_0g_1 \cdots g_n = e$, but I'm not sure...
The $1$-simplices in $BG$ have the form $* \xrightarrow{g} *$, so each one is a loop already. The set of all loops is therefore the set of all concatenations of these: the set of all paths. The homotopies will be determined by the fact that the path $* \xrightarrow{g} * \xrightarrow{h} *$ is homotopic to $* \xrightarrow{gh} *$ because there is a $2$-simplex with faces $g$, $h$, and $gh$. As a result, every loop will be homotopic to $* \xrightarrow{g} *$ for some $g$.