Connes showed in [Cohomologie cyclique et foncteurs $Ext^n$ (1983)](https://alainconnes.org/wp-content/uploads/n83.pdf) that the classifying space of his [cycle category](https://ncatlab.org/nlab/show/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 $\Lambda$?



Proofs that the classifying space of Connes' cycle category $\Lambda$ is $\mathbb C \mathbb P^\infty$
This statement appears as Corollary B.4 in Nikolaus-Scholze's [On topological cyclic homology](https://arxiv.org/abs/1707.01799); 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.


