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 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 (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.
This whole shift was particularly un-nerving for me because the most commonly described classifying spaces are infinite Grassmannians, which are generally understood in terms of colimit topologies. While this certainly works (since the tautological bundle is numerable and the total space is contractible), the lack of colimits in the general construction makes me feel like I've been thinking about things in the wrong way. As such, 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:
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. 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.
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.
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 $\text{colim}\,E_n^c$ (which agree for $G$ compact Hausdorff) 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? If it is not in fact possible to describe the universal bundle in this way, might it still be possible that $E_\infty^q/G$ or $(\text{colim}\,E_n^c)/G$ is a classifying space (i.e. homotopy equivalent, or weak homotopy equivalent, to $E_\infty^c/G$)? And if all of these approaches are doomed, why does the colimit explanation proliferate?
Addendum: In the second approach, I also have the following worry. I only see the proof of the cited fact 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)?
