I have had Milnor's construction of the classifying space of a topological group explained to me on multiple occasions, and seen it described briefly in various places. But only now am I reading the original papers of Milnor and Dold (or at least the appropriate sections) to learn the details of the constructions and classification.

I was startled to learn was that I was using the wrong topology on the join. Luckily, I have found some great explanations in §5.7 of Brown's *[Topology and Groupoids][1]* (for the topology on the join) and §14.4 of Dieck's *Algebraic Topology* (for the contractibility of the infinite join). However, I have been unable to reconcile these explanations with the less formal ones that I have seen/heard, which all centered on the classifying space as a colimit.

*What I seek here is an understanding of how (if it all) the colimit explanation can be factored into the general Milnor construction.* This requires some explanation of the hand-waving that I've heard, which may not be 100% accurate, so please bear with me. 

**Definition of the spaces:** The $n$-fold join $E_n=G*G*\dots$ (for any $1\leq n\leq \infty$) is a quotient of the set of all finite sequences $(t_1,g_1,\dots,t_m,g_m)$ with $m\leq n$ and $t_1+\dots+t_m=1$, where we identify $$(t_1,g_1,\dots,t_m,g_m)=(s_1,h_1,\dots,s_m,h_m)$$
if $t_i=s_i$ for all $i=1,\dots,m$ and $g_i=h_i$ whenever $t_i>0$. For each $i=1,\dots,n$, we have natural projection maps $t_i:E_n\rightarrow [0,1]$ and $g_i:t_i^{-1}(0,1]\rightarrow G$. There are two possible topologies that we can put on $E_n$. The coarse topology $E_n^c$ (adopting the terminology of Tyrone's comment) is the coarsest topology such that each of the projections $t_i$ and $g_i$ are continuous. The quotient topology $E_n^q$ is the topology induced on $E_n$ when we view it as a quotient of $G^n\times \Delta^n$ (where $G^\infty$ and $\Delta^\infty$ are both understood in terms of *finite* sequences). These topologies agree when $n<\infty$ and $G$ is compact Hausdorff, but in general, they are distinct. 

**The imprecise colimit explanation:** The colimit sketch that I have heard proceeds as follows (I will just write $E_n$, since the topology being used wasn't defined in these descriptions). Viewing $E_\infty$ as the colimit of the $E_n$'s, we can argue for either weak or full contractibility: 

1. The finite joins $E_n$ become increasingly connected as $n$ gets larger (this is true for both topologies, as mentioned in Milnor's "Construction of Universal Bundles, II"). If $G$ is Hausdorff, then points are closed in each $E_n$, so any map $S^k\rightarrow E_\infty$ must land in some finite $E_n$ by [this fact][2]. Since $\pi_k(E_n)=0$ for large $n$, we conclude that $E_\infty$ is weakly contractible. If $G$ is a CW-complex, then so is $E_\infty$, so we can actually conclude that $E_\infty$ is contractible.

2. For any $n<\infty$, the inclusion $E_n\subset E_n*G= E_{n+1}$ is null-homotopic (this is again true in both topologies).  Moreover, each of these inclusions is a cofibration (I still have to think about whether this is true in both topologies, but Tyrone has explained it in the comments for the coarse topology). The contractibility of $E_\infty$ then follows from [this fact][3].

**Issues with the colimit explanation:** For the colimit explanation to make any sense at all, we need to actually have $E_\infty=\text{colim}\,E_n$. This is true for the quotient topology, but I don't see why it is true for the coarse topology. Conversely, the description of $E_\infty$ as a principal numerable $G$-bundle is given for the coarse topology, and I am not sure if it holds for the quotient topology. 

There seem to be two possible avenues for a fix, but I am not sure that either works out. We could try and restrict to special cases where $E_\infty^c=\text{colim}\,E_n^c$, but Tyrone says that this is not necessarily true, even for $G$ compact Hausdorff. Alternatively, we could hope that $E_\infty^q$ or $\text{colim}\,E_n^c$ is still principal numberable, but I have doubts that this is true. 

*Is there some fix that makes the colimit argument work, even if only in certain special cases?* 

In the second approach, I also have the following worry. I only see the proof of the [cited fact][3] as holding in the category of pointed spaces, with basepoint-preserving maps and homotopies (otherwise, the point to which we are contracting seems like it could wander badly). This is an issue for the join, where the null-homotopy of $E_n\subset E_n*G=E_{n+1}$ slurps up $E_n$ to a point in $G$ in a non-basepoint-preserving fashion. Am I missing some argument that will make this still work (e.g. a contraction for basepoints not being preserved, or a null-homotopy of $E_n\subset E_{n+1}$ that preserves basepoints)?

  [1]: https://groupoids.org.uk/pdffiles/topgrpds-e.pdf
  [2]: https://math.stackexchange.com/questions/1584667/compact-subset-in-colimit-of-spaces
  [3]: https://mathoverflow.net/a/248449/147463