nLab straightening functor

This is a subentry of (∞,1)-Grothendieck construction.

Idea

When (∞,1)-categories are modeled as quasi-categories, the (∞,1)-Grothendieck construction takes the form of a Quillen equivalence between a model structure on sSet-enriched presheaves and a slice model structure of simplicial sets which has been introduced under the name straightening and unstraightening in Lurie 2009, Sec. 3.2:

Concretely, in the notation recalled at relation between quasi-categories and simplicial categories:

For a simplicial set $S$, simplicially enriched category $C$, and sSet-enriched functor $\mathfrak{C}[S] \to C$, there is a pair of adjoint functors

$$(St_\phi\dashv Un_\phi) \;\colon\; sSet/S \xrightarrow{\;\; St \;\;} [C^{op}, sSet]$$

which (under an assumption on the parameter $\phi$) can be shown to be a Quillen equivalence between

1. the slice category of sSet equipped with the model structure for right fibrations (also called contravariant model structure in HTT)

2. the category of simplicial presheaves equipped with global projective model structure.

There is also a Quillen equivalence

$$(St_\phi \dashv Un_\phi) \;\colon\; sSet^+/S \stackrel{St}{\;\; \longrightarrow \;\;} [C^{op}, sSet^+]$$

between

1. the model structure for Cartesian fibrations

2. the global projective model structure on functors with values in the model structure on marked simplicial sets.

In both cases,

• $St_\phi$ is called the straightening functor

• $Un_\phi$ is called the unstraightening functor.

These names have been chosen due to the fact that objects in the left hand category are defined by existential assertions and choices where on the right side these properties become coherence laws being part of the structure.

Related concepts

• cartesian fibration

• Grothendieck fibration

• (∞,1)-Grothendieck construction

• cleavage

References

The original result is due to:

• Jacob Lurie, Sec. 3.2 of: Higher Topos Theory, Annals of Mathematics Studies 170, Princeton University Press 2009 (pup:8957, pdf)

Intuition of Lurie’s construction is discussed in:

• MathOverflow: Why is the straightening functor the analogue of the Grothendieck construction?

Further discussion:

• Danny Stevenson, Covariant Model Structures and Simplicial Localization, North-West. Eur. J. Math. 3 (2017) 141-202 [arXiv:1512.04815]

• Gijs Heuts, Ieke Moerdijk, Left fibrations and homotopy colimits II [arXiv:1602.01274]

• Alexander Campbell, A modular proof of the straightening theorem, talk at Macquarie University (2020) [pdf, pdf]

• Fabian Hebestreit, Gijs Heuts, Jaco Ruit, A short proof of the straightening theorem (arXiv:2111.00069).

Discussion internal to any (∞,1)-topos:

• Louis Martini, Cocartesian fibrations and straightening internal to an ∞-topos [arXiv:2204.00295]

Discussion of straightening and unstraightening entirely within the context of quasi-categories:

• Denis-Charles Cisinski, Hoang Kim Nguyen, The universal coCartesian fibration [arXiv:2210.08945]

which (along the lines of the discussion of the universal left fibration from Cisinski 2019) allows to understand the universal coCartesian fibration as categorical semantics for the univalent type universe in directed homotopy type theory:

• Denis-Charles Cisinski, Hoang Kim Nguyen, Tashi Walde: Univalent Directed Type Theory, lecture series in the CMU Homotopy Type Theory Seminar (13, 20, 27 Mar 2023) [web, video 1:YT, 2:YT, 3:YT; slides 0:pdf, 1:pdf, 2:pdf, 3:pdf]

(see video 3 at 1:16:58 and slide 3.33).

Discussion of straightening and unstraightening for (∞,n)-categories:

• Lyne Moser, Nima Rasekh, Martina Rovelli, An (∞,n)-categorical straightening-unstraightening construction (arXiv:2307.07259).