Spectral sequences in \((\infty,1)\)-categories (Q2078386)

This paper sets up the homotopy spectral sequence of a simplicial or cosimplicial object in an \(\infty\)-category. Many of the most important spectral sequences fit into this context, including unstable Adams spectral sequences, the Eilenberg-Moore spectral sequence and the Serre spectral sequence. The homotopy spectral sequence computes the homotopy of the mapping object Map\(_X(y, \mathrm{colim}\, x)\), where \(x\) is a simplicial object and \(y\) a cogroup object in an \(\infty\)-category \(X\) with enough (co)limits. The construction via an exact couple is an \(\infty\)-categorical version of the spiral spectral sequence of [\textit{W. G. Dwyer} et al., J. Pure Appl. Algebra 103, No. 2, 167--188 (1995; Zbl 0860.55014)]. The paper explains how to construct the differentials of the spectral sequence in a model-invariant manner. They are described in terms of maps induced by the universal property of certain (homotopy) colimits and can be interpreted as obstructions to lifting certain diagrams from the homotopy category. The differentials are treated here as `relations', that is, as multi-valued functions defined on representatives in the \(E_1\)-page. An important ingredient is the Dwyer-Kan resolution of the restricted simplex category and an interesting feature is the role played by triangulations of permutahedra in describing the simplicial mapping spaces of the resolution.