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Controlled objects in left-exact $\infty$-categories and the Novikov conjecture

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arxiv 1911.02338 v3 pith:3CHGZNGG submitted 2019-11-06 math.KT math.ATmath.MG

Ulrich Bunke , Denis-Charles Cisinski , Daniel Kasprowski , Christoph Winges This is my paper · ORCID
classification math.KTmath.ATmath.MG
keywords left-exactinftyalgebraiccategoriescoarsecontrolledequivariantevery
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abstract

We associate to every $G$-bornological coarse space $X$ and every left-exact $\infty$-category with $G$-action a left-exact infinity-category of equivariant $X$-controlled objects. Postcomposing with algebraic K-theory leads to {new} equivariant coarse homology theories. This allows us to apply the injectivity results for assembly maps by Bunke, Engel, Kasprowski and Winges to the algebraic K-theory of left-exact $\infty$-categories.

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Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Branched coarse coverings and transfer maps

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    A new transfer formalism for coarse K-homology theories yields an operator-free proof of Atiyah's L2-index theorem and a fresh treatment of Higson's counterexample to the coarse Baum-Connes conjecture.

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    For Q-schemes and Dedekind schemes, the motivic t-structure exists on 1-motives with integral coefficients, with heart the abelian category of Deligne 1-motives with torsion.

  3. Higher $K$-theory of forms III: from chain complexes to derived categories

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    Hermitian K-theory of an exact form category is canonically equivalent to the hermitian K-theory of its derived Poincaré ∞-category, with no assumption that 2 is invertible.

  4. Coarse cone quotients

    math.AT 2025-07 conditional novelty 6.0 of 10

    For uniformly free, ergodic actions with spectral gap, the motivic coarse assembly map for the cone quotient O∞(X)//G fails to be an equivalence.

  5. Finite asymptotic dimension and the coarse assembly map

    math.AT 2024-12 conditional novelty 6.0 of 10

    The coarse assembly map for strong coarse homology theories with weak transfers is a phantom equivalence for bornological coarse spaces of weakly finite homotopical asymptotic dimension.

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