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Derived $\infty$-categories as exact completions

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arxiv 2310.12925 v1 pith:2APVNHM4 submitted 2023-10-19 math.CT math.AT

Germán Stefanich This is my paper · ORCID
classification math.CTmath.AT
keywords inftyexactcategoriescategoryabeliancategoricalcompletionsderived
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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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Cited by 3 Pith papers

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  1. Dualizable Additive Categories

    math.AT 2026-08 conditional novelty 8.0 of 10

    Dualizable additive categories are characterized intrinsically and via almost modules, and a universal finitary localizing invariant (prestable motives) is constructed whose unit corepresents algebraic K-theory.

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    math.CT 2026-08 accept novelty 8.0 of 10

    Every n-cotorsion pair on a triangulated category has a heart that is an abelian n-truncated category, carrying compatible pretriangulated and extriangulated structures.

  3. A family completion theorem for tempered cohomology

    math.AT 2026-08 accept novelty 7.0 of 10

    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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