Cellular $E_k$-algebras

Abstract

We give a set of foundations for cellular E k -algebras which are especially convenient for applications to homological stability. We provide conceptual and computational tools in this setting, such as filtrations, a homology theory for E k -algebras with a Hurewicz theorem, CW approximations, and many spectral sequences, which shall be used for such applications in future papers.

Stability Properties of Moduli Spaces

“…Recently Galatius, Kupers and Randal-Williams [GKRW18] identified and proved a new kind of stabilization result, which they describe by the slogan "the failure of homological stability is itself stable". They defined homological-degree-shifting stabilization maps and use them to prove secondary homological stability for the homology of mapping class groups and general linear groups outside the stable range of (primary) homological stability.…”

Exposé Bourbaki 1164 : Homology of Hurwitz spaces and the Cohen-Lenstra heuristic for function fields (after Ellenberg, Venkatesh, and Westerland)

“…(1) Though Galatius, Kupers, and I have recently proposed another [GKRW18a], which in fortunate circumstances can provide information beyond classical homological stability, for example in the case of mapping class groups of surfaces [GKRW18b].…”

“…In this exposition of [EVW16] I will present their argument differently to the way it appears in that paper, closer to the framework of [GKRW18a] than to the classical approach to homological stability described above. While many of the key steps are unchanged, I find that this streamlined argument clarifies the essential points.…”

Homotopy pro-nilpotent structured ring spectra and topological Quillen localization

“…TQ-homology and its relative form, topological Andre-Quillen (TAQ) homology, first introduced in [4] for commutative ring spectra (see also [3,5,6,21,35,38,39,45]), are defined as derived indecomposables of O-algebras analogous to Quillen homology of commutative algebras [1,44]; see also [19,24,20,41]. More precisely, factoring the canonical truncation map O → τ 1 O (see [28]) in the category of operads as O → J → τ 1 O, a cofibration followed by a weak equivalence, we get the corresponding change of operads adjunction…”