Total cofibres of diagrams of spectra (Q2581079)

Fix a pair \(\mathcal{D}\subset \mathcal{C}\) of posets, considered as categories. Define the total cofiber \(\Gamma(X)\) of a diagram \(X:\mathcal{C}\to\mathcal{SS}ets_\ast\) of pointed simplicial sets as the strict cofiber of the map \(\text{hocolim}_{\mathcal D}(X)\to \text{hocolim}_{\mathcal C}(X)\). Analogously, the total cofiber of a diagram of pointed topological spaces is defined. By applying the definition levelwise, we obtain a total cofiber functor for diagrams of spectra in the sense of \textit{A. K. Bousfield} and \textit{E. M. Friedlander} [Geom. Appl. Homotopy Theory, II, Proc. Conf., Evanston 1977, Lect. Notes Math. 658, 80-130 (1978; Zbl 0405.55021)]. The total cofiber functor has been used by the author in his studies of quasi-coherent sheaves on projective toric varieties as a replacement of the global section functor and its derived functors. If \(\mathcal{C}\) is the poset of non-empty faces of a polytope and \(\mathcal{D}\) the subposet of proper faces and \(X\) is a \(\mathcal{C}\)-diagram of pointed spaces the author constructed a spectral sequence \[ E^2_{p,q}= \lim{}^{n-p}\widetilde{H}_q(X;\mathbb{Z})\to \widetilde{H}_{p+q}(\Gamma(X);\mathbb{Z}) \] where \(n\) is the dimension of the polytope [\textit{T. Hüttemann}, K-Theory 31, 101--123 (2004; Zbl 1068.55015)]. This spectral sequence can be interpreted as a device for comparing the homotopy limit of \(X\) with the total cofiber. The present paper investigates this relationship more closely. For a diagram \(X\) of pointed spectra the author constructs a map \[ \widetilde{\Gamma}_X:\text{holim}_{\mathcal C}X\to \Hom_{\mathcal{SS}ets_\ast} (\Gamma(N(\mathcal{C}\downarrow -)_+), \widetilde{\Gamma}(X)) \] from the homotopy limit into a mapping space and characterizes all pairs of posets \(\mathcal{D}\subset\mathcal{C}\) for which \(\widetilde{\Gamma}_Y\) is a stable weak equivalence of spectra. Here \(N\) is the nerve functor and \(\tilde{\Gamma}(Y)\) is the levelwise fibrant replacement of \(\Gamma(X)\). As an application he extends the spectral sequence above to more general diagrams and derives homotopy spectral sequences for the total cofiber.