Topological Hochschild homology and the cyclic bar construction in symmetric spectra (Q2814428)

Let \(R\) and \(M\) be an associative symmetric ring spectrum and an \(R\)-bimodule, respectively. One way of defining Topological Hochschild Homology (THH) is as the homotopy colimit \(\mathrm{hocolim}_{\mathcal{I}^{\times k+1}}\mathcal{D}_k(R;M)\) where \(\mathcal{D}(R;M)([k];-):\mathcal{I}^{\times k+1}\to Sp^{\Sigma}\) is a suitable functor from the \((k+1)\)-power of the category of finite sets to the category of symmetric spectra of simplicial sets. On the other hand, there is the \textit{twisted cyclic bar construction} \(B^{cy}_{\bullet}(R;M)\). By Theorem 4.2.8 in [\textit{B. Shipley}, \(K\)-Theory 19, No. 2, 155--183 (2000; Zbl 0938.55017)], there is a zig-zag of stable equivalences of simplicial objects that relates these constructions that induce a chain of stable equivalences after realization. In this paper, the authors correct an error in Shipley's proof of this theorem. The error is an incompatibility condition of the morphism in degrees 0 and 1.