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.

[3]André Haefliger: Homotopy and Integrability
[4]Tammo tom Dieck: On The Homotopy Type of Classifying Space

  • $\begingroup$ I don't know the answer to your question. But in the remarks of the paper, on page 310 before the appendix, Segal notes that throughout the paper he always means realization to be the realization after replacing the diagram by one where the degeneracies are closed Hurewicz cofibrations. $\endgroup$ – Matthew Sartwell Nov 11 '15 at 23:59
  • $\begingroup$ @Matthew: (Hi, nice to see you here!) Yes, indeed. But he also referred to his article 'Classifying Space of Spectral Sequence' in the beginning of the sec.2 when he mentioned the nerve of $\mathcal{C}$. And in that article the realization is the thin one. So I assume he used the good resolution only after passing from $\Gamma$-category to $\Gamma$-space. $\endgroup$ – yisheng Nov 12 '15 at 15:50

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