Unstraightening for Segal spaces

AG Seminar, January 27rd 2022: - Joost Nuiten

Unstraightening for Segal spaces

The straightening-unstraightening correspondence of Lurie provides an equivalence between cocartesian fibrations over an infinity-category and functors with values in infinity-categories. In this talk, I will describe an alternative proof of this equivalence in terms of Segal spaces. This proof is closer in spirit to the classical Grothendieck construction and relies on a combinatorial result relating two types of fibrations between double categories. If time permits, I will sketch how an iterated application of this result leads to a version of unstraightening for higher categories.