Let $G$ be a topological group and consider a family $$G\rightarrow E_i\rightarrow B_i$$ of $G$-principal bundles indexed over the natural numbers. Suppose we have $G$-bundle morphisms (equivariant bundle maps) $$\require{AMScd}\begin{CD} G @>{=}>> G\\ @VVV @VVV \\ E_i @>{f_i}>> E_{i+1}\\ @VVV @VVV \\ B_i @>{g_i}>> B_{i+1}\\ \end{CD},$$

**Is the colimit $$\operatorname{colim}_{i} E_i\rightarrow \operatorname{colim}_{i} B_i$$ always a $G$-principal bundle if one does not assume any topological restrictions?****Are there topological restrictions, such that 1) is true? Maybe some, which include all the examples I gave? A possible option would be to impose the maps $E_i\rightarrow E_{i+1}$ to be (closed) inclusion. This should be the case for all examples given.****Does it hold it the category of compactly generated Hausdorff spaces?****Does it hold in the category of compactly generated weak Hausdorff spaces?**

This situation occurs quite often in algebraic topology, especially to find models for classifying spaces:

- $S^0\rightarrow S^{(n+1)-1}\rightarrow \mathbb{R}P^n$ yields $S^0\rightarrow S^\infty\rightarrow \mathbb{R}P^\infty$ and therefore $BS^0\simeq \mathbb{R}P^\infty$.
- $S^1\rightarrow S^{2(n+1)-1}\rightarrow \mathbb{C}P^n$ yields $S^1\rightarrow S^\infty\rightarrow \mathbb{R}P^\infty$ and therefore $BS^1\simeq \mathbb{C}P^\infty$.
- $S^3\rightarrow S^{4(n+1)-1}\rightarrow \mathbb{H}P^n$ yields $S^3\rightarrow S^\infty\rightarrow \mathbb{H}P^\infty$ and therefore $BS^3\simeq \mathbb{H}P^\infty$.
- $\operatorname{Diff}(M)\rightarrow \operatorname{Emb}(M,\mathbb{R^n})\rightarrow \operatorname{Emb}(M,\mathbb{R^n})/\operatorname{Diff}(M)$ yields $\operatorname{Diff}(M)\rightarrow \operatorname{Emb}(M,\mathbb{R^\infty})\rightarrow \operatorname{Emb}(M,\mathbb{R^\infty})/\operatorname{Diff}(M)$ and therefore $B\operatorname{Diff}(M)\simeq \operatorname{Emb}(M,\mathbb{R^\infty})/\operatorname{Diff}(M)$ for a compact smooth manifold.

I want to put these examples in a general setting. To conclude statements about the classifying space, one needs to know that the colimit is weakly contractible, which is mostly implied by the increasing connectivity of the spaces $E_i$. For sure, one has to impose topological restrictions to conclude, that the filtered colimit commutes with homotopy groups, e.g. that the spaces $E_i$ are compactly generated spaces and the maps $f_i\colon E_i\rightarrow E_{i+1}$ are closed inclusions.

**Did I say anything wrong?**