Condensed Group Cohomology

TL;DR

This paper explores condensed group cohomology, showing it refines continuous cohomology for broad classes of topological groups and relates it to sheaf and classifying space cohomologies.

Contribution

It establishes that for many topological groups, continuous group cohomology with solid coefficients can be realized as a derived functor in the condensed setting.

Findings

Condensed group cohomology refines continuous cohomology.

Continuous cohomology with solid coefficients can be realized as a condensed derived functor.

Generalizes identifications of condensed with sheaf cohomology.

Abstract

Condensed mathematics as developed by Clausen and Scholze yields a version of derived functors over the category of continuous $G$-modules for a Hausdorff topological group $G$. We study the resulting notion of group cohomology and its relation to continuous group cohomology and the condensed/sheaf/singular cohomology of classifying spaces. While condensed group cohomology is generally a more refined invariant than continuous group cohomology, we show that for a broad class of topological groups, continuous group cohomology with solid coefficients, such as locally profinite continuous $G$-modules, can be realized as a derived functor in the condensed setting. We also revisit cornerstones of condensed mathematics, paying special attention to set-theoretic size issues. To this end, we review a framework for working with accessible (hyper)sheaves on large sites satisfying suitable accessibility conditions and show that the associated categories retain many topos-like properties. Moreover, we generalize identifications of condensed with sheaf cohomology obtained by Clausen and Scholze.