Condensed Group Cohomology

TL;DR

The paper develops condensed group cohomology as Ext-groups in the abelian category of condensed modules, relating it to continuous cohomology and to the (singular) and sheaf cohomology of classifying spaces BG. It identifies cases where continuous cohomology with solid/locally profinite coefficients matches the condensed derived functor, and explains how condensed cohomology of BG aligns with familiar cohomology when BG is a numerable classifying space. Beyond concrete comparisons, the work builds a comprehensive topos-theoretic and categorical framework for large sites: accessible (hyper)sheaves, big presentable categories, and spectrum-valued cohomology, together with solid/condensed module theory, to systematize cohomology in condensed settings. The results yield both comparisons to classical invariants and a robust, flexible platform to study cohomology in large, topos-like contexts, including extensions to solid and condensed algebraic structures and to classifying spaces.

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.

Theorems & Definitions (738)

• Theorem 1: \ref{['condensedcohomologyiscohomologyofclassifyingspaces']}, \ref{['condensedgroupcohomologyissheafcohomologyofbg']}
• Theorem 2: \ref{['solidequalscontinuousgoodgroups']}
• Theorem 3: \ref{['condensedandsheafcohomology1']}
• Theorem 4: \ref{['condensedandsheafcohomologylocallycontractible']}
• Theorem 5
• Lemma 6: \ref{['universalpropertyspectrumobjectsbigpresentablemonoidal']}
• Lemma 7: \ref{['internalhommodules']}
• Lemma 8: \ref{['condensedmodulesarederivedcats']}
• Remark
• Proposition : Scholzecondensed, \ref{['kappacontinuousfullyfaithfullyintocondensed']}, \ref{['section:condensedsetsandtopologicalspaces']}