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 $mathbb Cmathbb P^infty$ 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 $Lambda$?