Summary: The category of topological abelian groups is, despite the name of its objects, not abelian. The problem is, for instance, that the continuous homomorphism from the real numbers with discrete topology to the real numbers with usual topology is not an isomorphism despite having zero kernel and cokernel. As a corollary, there was no natural definition of the bounded derived category of locally compact abelian groups. The goal of this seminar is to present the solution to this problem developed by Scholze and Clausen: the theory of condensed sets, which allows us to do homological algebra with topological groups or rings without thinking about topology.
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