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Presentable $(\infty, n)$-categories

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arxiv 2011.03035 v1 pith:BQJBEJCP submitted 2020-11-05 math.AT math.CT

classification math.ATmath.CT
keywords inftycategoriesconicalmathcalpresentablecategorycolimitsmathrm
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abstract

We define for each $n \geq 1$ a symmetric monoidal $(\infty, n+1)$-category $n\mathrm{Pr}^L$ whose objects we call presentable $(\infty,n)$-categories, generalizing the usual theory of presentable $(\infty,1)$-categories. We show that each object $\mathcal{C}$ in $n\mathrm{Pr}^L$ has an underlying $(\infty,n)$-category $\psi_n(\mathcal{C})$ which admits all conical colimits, and that conical colimits of right adjointable diagrams in $\psi_n(\mathcal{C})$ can be computed in terms of conical limits after passage to right adjoints.

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Cited by 4 Pith papers

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

  1. Enriched $\infty$-categories as marked module categories

    math.AT 2025-01 conditional novelty 8.0 of 10

    Enriched ∞-categories are equivalent to presentable module categories marked by an atomically generating family of representables.

  2. On The Telescopic Picard Group

    math.AT 2024-12 conditional novelty 8.0 of 10

    For all primes p and heights n, Pic(Sp_{T(n)}) contains Z_p × Z/(a_p(p^n−1)), lifting the known K(n)-local subgroup.

  3. Higher Semiadditive Character Theory

    math.AT 2026-07 accept novelty 7.0 of 10

    Every ∞-commutative monoid has a universal (n−t)-fold semiadditive character that blue-shifts height, recovers the transchromatic character on Morava E-theory, and computes L_Q(S^A_{K(n)}) via GL_{n−t}(Z_p)-fixed points.

  4. Enriched $\infty$-operads as marked algebras

    math.AT 2026-07 accept novelty 7.0 of 10

    A V-enriched ∞-operad is equivalent to a presentably symmetric monoidal V-module category generated by a ⊗-atomic marking of its colors.

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