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Monoidal envelopes and Grothendieck construction for dendroidal Segal objects

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arxiv 2301.10751 v1 pith:HF6WDGJY submitted 2023-01-25 math.CT math.AT

classification math.CTmath.AT
keywords dendroidalconstructionsegalcategoryenvelopesfibrationsmonoidalspaces
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abstract

We propose a construction of the monoidal envelope of $\infty$-operads in the model of Segal dendroidal spaces, and use it to define cocartesian fibrations of such. We achieve this by viewing the dendroidal category as a "plus construction" of the category of pointed finite sets, and work in the more general language of algebraic patterns for Segal conditions. Finally, we rephrase Lurie's definition of cartesian structures as exhibiting the categorical fibrations coming from envelopes, and deduce a straightening/unstraightening equivalence for dendroidal spaces.

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Cited by 2 Pith papers

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

  1. Day convolution for algebraic patterns

    math.CT 2026-03 accept novelty 6.0 of 10

    For a robust algebraic pattern, exponentiable weak Segal fibrations are exactly those satisfying a Conduché-style factorization condition.

  2. 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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