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On straightening for Segal spaces
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abstract
The straightening-unstraightening correspondence of Grothendieck--Lurie provides an equivalence between cocartesian fibrations between $(\infty, 1)$-categories and diagrams of $(\infty, 1)$-categories. We provide an alternative proof of this correspondence, as well as an extension of straightening-unstraightening to all higher categorical dimensions. This is based on an explicit combinatorial result relating two types of fibrations between double categories, which can be applied inductively to construct the straightening of a cocartesian fibration between higher categories.
Forward citations
Cited by 2 Pith papers
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Colimits in Oriented Category Theory
Oriented colimits generalize lax colimits and the Gray tensor product and yield a Gray-enriched straightening equivalence between presheaves and cocartesian fibrations of (∞,∞)-categories.
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Fibrations in Oriented Category Theory
The paper establishes several equivalent characterizations of fibrations of (∞,∞)-categories, shows the category of fibrations forms an oriented category, and constructs free, universal, and Grothendieck-construction ...
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