On straightening for Segal spaces

Abstract

We study cocartesian fibrations between (∞, d)-categories in terms of higher Segal spaces and prove a version of straightening for them. This follows from a repeated application of an explicit combinatorial result, relating two types of fibrations between double categories.

Cocartesian fibrations and straightening internal to an $\infty$-topos

“…In the special case where B is the ∞-topos of spaces, Theorem 6.3.1 recovers Lurie's straightening equivalence. One can therefore regard our proof of Theorem 6.3.1 as another approach to the straightening equivalence, complementing existing proofs such as Lurie's original account in [Lur09] and the more recent approaches by Boavida de Brito [BdB18], Nuiten [Nui21], Hebestreit-Heuts-Ruit [HHR21] and Rasekh [Ras21].…”

“…We have already mentioned above that by now there exist several proofs for the ∞-categorical straightening equivalence [Lur09,BdB18,Nui21,HHR21,Ras21]. In [Sha18], Jay Shah builds upon this result to derive a straightening equivalence for parametrised higher categories.…”

What is an equivalence in a higher category?

“…In a similar vein, given any prime number 𝑝, there's an (∞, 1)-equivalence between the ∞-category of 𝑝-adic spaces and an appropriate ∞-category of 𝔼 ∞ -F𝑝 -algebras. (2) 2-, (∞, 1)and (∞, 𝑛)-categorical straightening-unstraightening [44,80,89,90]: Given a 2category , there is a 2-equivalence of 2-categories between the 2-category of categorical pseudo-presheaves over  and the 2-category of Grothendieck fibrations over . Also, given a ∞-(or more generally (∞, 𝑛)-)category , there is an (∞, 1)-equivalence between the ∞category of ∞-(or more generally (∞, 𝑛)-)categorical presheaves and the ∞-category of cartesian fibrations over .…”