I am reading Segal's paper '**categories and cohomology theories**' [1], but there is one claim (in the last example in sec.2) I don't quite understand:

Let $\mathcal{C}$ be the category of bounded chain complex over $\mathbb{R}$ ,with quasi-isomorphism as morphisms, and we topologize them as subspaces of $\mathbb{R}^{N}$, for some large $N\in\mathbb{N}$.

More precisely, Let $K_{n}$ be the space of the chain complex $E_{\ast}$ such that $E_{i}=\mathbb{R}^{n_{i}}$, where $n=\{n_{i}\}_{i\in\mathbb{Z}}$. We topologize it as a subspace of some copies of $\mathbb{R}$ by $d_{i}d_{i+1}=0$, then define $$ob(\mathcal{C})=\coprod_{n}K_{n}$$, similarly we can topologize $mor(\mathcal{C})$ as subspace of some copies of $\mathbb{R}$, namely by $f_{i}d_{i+1}=d_{i+1}f_{i+1}$.

Then the claim is $\vert \mathcal{C}\vert$ is a classifying space for real vector bundles over the compact spaces.

My approach is to use Haefliger's classifying space of topological groupoid to identify $[X,\vert\mathcal{C}\vert]$ with the $\mathcal{C}$-structure over $X$, a compact space, and then use these $\mathcal{C}$-cocycles over $X$ to connect to the model given by Segal in his another article '**Equivariant K-theory**'[2](in the appendix).

The problems I encounter are:

$\mathcal{C}$ is not really the topological groupoid that I know about

(though it is called so in [1]), therefore Haefliger's theorem [3] cannot be applied directly in this case.

Moreover, as far as I understand, in that example $\vert\mathcal{C}\vert$ means the thin realization. But it is not always equivalent to the model used by Haefliger.

P.S. The Haefliger's model is a generalization of Milnor's join construction which, if I understand correctly, is equivalent to the fat realization [4], not the thin one, in general.

[1]http://www.sciencedirect.com/science/article/pii/0040938374900226

[2]http://link.springer.com/article/10.1007%2FBF02684593#page-1

[3]André Haefliger: Homotopy and Integrability

[4]Tammo tom Dieck: On The Homotopy Type of Classifying Space