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Connes showed in Cohomologie cyclique et foncteurs $Ext^n$ (1983) that the classifying space of his cycle category $\Lambda$ is $\mathbb C \mathbb P^\infty = B(S^1) = K(\mathbb Z,2)$.

Connes' proof is not quite as conceptual as one might like. He shows that $|\Lambda|$ is simply-connected, and then computes its cohomology via an explicit resolution of the constant functor $\Lambda \to Ab$, $[n] \mapsto \mathbb Z$ via a double complex which is cooked-up by gluing together the usual way of resolving cyclic groups and the usual way of resolving simplicial objects, with a few modifications. He's able to get the ring structure on the cohomology using an endomorphism of this double complex. The result follows because $\mathbb C\mathbb P^\infty$ is characterized among simply-connected spaces by its cohomology ring.

Surely in the last nearly 40 years new proofs that $|\Lambda| \simeq \mathbb C \mathbb P^\infty$ have been found.

Question 1: What are some alternate proofs that $|\Lambda| \simeq \mathbb C \mathbb P^\infty$?

Question 2: (somewhat vague) In particular, is there a proof which somehow uses an explicit $S^1$ action on some category or simplicial set related to $\Lambda$?

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    $\begingroup$ Cisinski gives a nice proof of this in Proposition 8.5.19 of his Les préfaisceaux comme modèles des types d'homotopie. $\endgroup$
    – Andrea Gagna
    Commented Apr 16, 2021 at 21:03
  • $\begingroup$ Oh that's fantastic -- it looks a lot like the proof in Nikolaus and Scholze but of course Cisinski predates Nikolaus and Scholze considerably. Cisinski, like the nlab, denotes the paracyclic category by "$L$". $\endgroup$
    – Tim Campion
    Commented Apr 16, 2021 at 21:07
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    $\begingroup$ It's also in Hoyois' paper on the homotopy fixed points of the circle action on HH (corollary 1.2) $\endgroup$
    – Maxime Ramzi
    Commented Apr 16, 2021 at 21:44

2 Answers 2

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This statement appears as Corollary B.4 in Nikolaus-Scholze's On topological cyclic homology; essentially, $\Lambda$ is the quotient of the paracyclic category $\Lambda_\infty$ by a free $B\mathbb Z$-action, which receives a final map from the simplicial category, hence is contractible.

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  • $\begingroup$ Thanks, this is great. I wasn't quite sure where to start in the vast literature on cyclic things. The first few references I tried just referred to Connes for the computation of the homotopy type. $\endgroup$
    – Tim Campion
    Commented Apr 16, 2021 at 21:10
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    $\begingroup$ Incidentally, I think there's a slight error in Nikolaus and Scholze's proof in Theorem B.3 that $\Delta \to \Lambda_\infty$ is a final functor. The criterion they formulate for contractibility does not apply because some of the finite intersections of $U_t$'s are empty rather than contractible. This is easily remedied by inductively showing that $\mathcal C[-n,n)$ is contractible by induction on $n$, using a similar glueing argument to their criterion, and passing to the colimit. $\endgroup$
    – Tim Campion
    Commented Apr 16, 2021 at 22:03
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Another proof appears as Lemma 8.1.1.15 and Remark 8.1.1.16 of Igusa's Higher Franz-Reidemeister torsion. It uses spaces of graphs to verify that $|\Lambda|$ classifies oriented circle bundles.

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