ScalingStacks

A.4 Straightening and unstraightening[00IK]

A fundamental tool in \(\infty\)-category theory is the straightening and unstraightening equivalence, as we now briefly describe. ⁴⁹

Fix an \(\infty\)-category \(\mathcal B\) and a functor \(\mathcal B\xrightarrow{F} \mathrm{Cat}_\infty\). Then, the (coCartesian) unstraightening of \(F\) (a.k.a. its (covariant) Grothendieck construction) is an object \((\mathcal E\xrightarrow{p} \mathcal B) \in (\mathrm{Cat}_\infty)_{/\mathcal B}\) that may be described heuristically as follows:

• an object of \(\mathcal E\) is given by a pair of an object \(b \in \mathcal B\) and an object \(x \in F(b)\);

• a morphism \((b,x) \rightarrow(c,y)\) in \(\mathcal E\) is given by a morphism \(b \xrightarrow{f} c\) in \(\mathcal B\) along with a morphism \(F(f)(x) \xrightarrow{\alpha} y\) in \(F(c)\).

(Of course, the images under \(p\) of these data are simply \(b\) and \(f\), respectively.) Such a morphism \((f,\alpha)\) in \(\mathcal E\) is called (\(p\)-)coCartesian if \(\alpha\) is an equivalence. Observe that these satisfy a universal property: if \(e \xrightarrow{\varphi} f\) in \(\mathcal E\) is \(p\)-coCartesian, then for any \(g \in \mathcal E_{p(f)}\) we have an equivalence \(\mathrm{Hom}_\mathcal E(f,g) \simeq \mathrm{Hom}_\mathcal E(e,g) \times_{\mathrm{Hom}_\mathcal B(p(e),p(g))} \{p(\varphi) \}\).

Conversely, a functor \(\mathcal E\xrightarrow{p} \mathcal B\) is called a coCartesian fibration if for every pair of an object \(e \in \mathcal E\) and a morphism \(p(e) \xrightarrow{f} b\) in \(\mathcal B\), the morphism \(f\) admits a coCartesian lift with source \(e\). In this case, \(p\) is the unstraightening of a functor \(\mathcal B\xrightarrow{F} \mathrm{Cat}_\infty\), whose values are given by the fibers \(F(b) \simeq \mathcal E_b\) and whose functoriality is implicitly specified by the coCartesian morphisms (in essence because the Yoneda embedding is fully faithful). We refer to \(F\) as the straightening of \(p\), and to its functoriality \(F(b) \xrightarrow{F(f)} F(c)\) for a morphism \(b \xrightarrow{f} c\) in \(\mathcal B\) as the coCartesian monodromy functor of \(\mathcal E\) associated to \(f\).

The coCartesian fibrations over \(\mathcal B\) define a (generally non-full) subcategory \({\textup{coCart}}_\mathcal B\subseteq (\mathrm{Cat}_\infty)_{/\mathcal B}\), whose morphisms are those functors over \(\mathcal B\) that preserve coCartesian morphisms. Altogether, by [Lur09, Thm. 3.2.0.1], straightening and unstraightening define inverse equivalences \[\mathrm{Fun}(\mathcal B,\mathrm{Cat}_\infty) \simeq {\textup{coCart}}_\mathcal B ~,\] under which precomposition with a functor \(\mathcal B' \rightarrow\mathcal B\) corresponds to pullback therealong.

A similar but dual story applies in the case of a functor \(\mathcal B^\mathrm{op}\xrightarrow{F} \mathrm{Cat}_\infty\): this now has a (Cartesian) unstraightening (a.k.a. its (contravariant) Grothendieck construction), giving an object \((\mathcal E\rightarrow\mathcal B) \in (\mathrm{Cat}_\infty)_{/\mathcal B}\) admitting a dual description. Altogether, we obtain an analogous equivalence \[\mathrm{Fun}(\mathcal B^\mathrm{op},\mathrm{Cat}_\infty) \simeq {\textup{Cart}}_\mathcal B ~.\]