ScalingStacks

A.5 Adjunctions revisited[00IL]

An adjunction of \(\infty\)-categories can be defined as a functor \(\mathcal E\rightarrow[1]\) that is both a coCartesian fibration and a Cartesian fibration. Its coCartesian unstraightening defines the left adjoint \(\mathcal E_0 \xrightarrow{L} \mathcal E_1\), while its Cartesian unstraightening defines the right adjoint \(\mathcal E_0 \xleftarrow{R} \mathcal E_1\), and the universal properties of coCartesian and Cartesian morphisms yield natural equivalences \(\mathrm{Hom}_{\mathcal E_0}(e,R(f)) \simeq \mathrm{Hom}_\mathcal E(e,f) \simeq \mathrm{Hom}_{\mathcal E_1}(L(e),f)\) for any \(e \in \mathcal E_0\) and \(f \in \mathcal E_1\).

We define a morphism of adjunctions to be a morphism in \({\textup{coCart}}_{[1]} \cap {\textup{Cart}}_{[1]}\).⁵⁰ In particular, a morphism of adjunctions determines a commutative square in \(\mathrm{Cat}_\infty\) after omitting either both left adjoints or both right adjoints.

Given a commutative square in \(\mathrm{Cat}_\infty\) in which two parallel functors are both (say) left adjoints, passing to their right adjoints we obtain a canonical laxly-commutative square (i.e. one that commutes up to a specified natural transformation), and it is merely a condition for this to be invertible so that the original square defines a morphism adjunctions [Hau+23].⁵¹ Of course, this is nothing but the condition that the morphism in \({\textup{coCart}}_{[1]}\) specified by the original square lies in the subcategory \({\textup{coCart}}_{[1]} \cap {\textup{Cart}}_{[1]}\). Dual remarks apply if the two parallel functors are instead both right adjoints.