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Derived $\infty$-categories as exact completions

Germán Stefanich · 2023 · math.CT · DOI 10.48550/arxiv.2310.12925 · arXiv 2310.12925

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abstract

We develop the theory of exact completions of regular $\infty$-categories, and show that the $\infty$-categorical exact completion (resp. hypercompletion) of an abelian category recovers the connective half of its bounded (resp. unbounded) derived $\infty$-category. Along the way, we prove that a finitely complete $\infty$-category is exact and additive if and only if it is prestable, extending a classical characterization of abelian categories. We also establish $\infty$-categorical versions of Barr's embedding theorem and Makkai's image theorem.

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A family completion theorem for tempered cohomology

math.AT · 2026-08-03 · accept · novelty 7.0

For oriented P-divisible groups over noetherian E-infinity rings, family-completion of tempered cohomology modules is equivalent to algebraic completion at the corresponding ideal, generalizing Atiyah–Segal and AHJM.

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  • A family completion theorem for tempered cohomology math.AT · 2026-08-03 · accept · none · ref 111 · internal anchor

    For oriented P-divisible groups over noetherian E-infinity rings, family-completion of tempered cohomology modules is equivalent to algebraic completion at the corresponding ideal, generalizing Atiyah–Segal and AHJM.