Year: 2025
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Brink, Emma
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4417030997
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