In his beautiful paper *On the cohomology and K theory of the general linear group over a finite field*, Quillen constructs (if I understand correctly) an isomorphism on connected components of K-theory $K(\mathbb{F}_q)\mid_{\ge 1} \cong \text{Fib}(1-\Psi^q:ku\to ku).$ His construction, while very cool, seems to be based on a miracle: that the Brauer lift produces complex representations from mod-$p$ ones. It also uses the +-construction for defining K-theory (though perhaps it's not that hard to update his construction to work with Segal formalism)?

I'm curious whether there is a more modern perspective on this equivalence of spectra, maybe using cyclic structure on topological Hochschild homology.