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Algebras over infinity-operads

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arxiv 1110.1776 v2 pith:N43ZJ5AC submitted 2011-10-08 math.AT math.CT

Gijs Heuts This is my paper · ORCID
classification math.ATmath.CT
keywords algebrascocartesiandendroidalinfinity-categorysetscocartdefinitionequivalence
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We develop a notion of an algebra over an infinity-operad with values in infinity-categories which is completely intrinsic to the formalism of dendroidal sets. Its definition involves the notion of a coCartesian fibration of dendroidal sets and extends Lurie's definition of a coCartesian fibration of simplicial sets. We show how, for a dendroidal set X, the coCartesian fibrations over X fit together to form an infinity-category coCart(X). Using a generalization of the Grothendieck construction, we prove that coCart(X) is equivalent to the infinity-category of algebras in infinity-categories over the simplicial operad associated to X. This equivalence can be restricted to give an equivalence between algebras taking values in infinity-groupoids (or equivalently, spaces) and the infinity-category of so-called left fibrations over X.

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Cited by 3 Pith papers

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

  1. Colimits in Oriented Category Theory

    math.AT 2026-08 conditional novelty 7.0 of 10

    Oriented colimits generalize lax colimits and the Gray tensor product and yield a Gray-enriched straightening equivalence between presheaves and cocartesian fibrations of (∞,∞)-categories.

  2. The root functor

    math.AT 2025-05 conditional novelty 7.0 of 10

    Every normal dendroidal ∞-operad is an operadic weak equivalence to the localization at root-preserving maps of the nerve of its discrete, Σ-free operad of elements.

  3. A straightening-unstraightening equivalence for $\infty$-operads

    math.AT 2025-01 accept novelty 5.0 of 10

    For any Lurie infinity-operad, the infinity-category of operadic left fibrations is equivalent to the infinity-category of algebras in spaces.

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