ScalingStacks

A.2.3 Adjunctions[00ID]

It is merely a condition for a functor \(\mathcal C\xrightarrow{F} \mathcal D\) to be a (say) left adjoint: its space of right adjoints is either empty or contractible. First of all, a pointwise right adjoint to \(F\) at an object \(d \in \mathcal D\) is a pair of an object \(c \in \mathcal C\) and a morphism \(F(c) \xrightarrow{\varepsilon_d} d\) such that for every \(c' \in \mathcal C\) the composite \(\mathrm{Hom}_\mathcal C(c',c) \xrightarrow{F} \mathrm{Hom}_\mathcal D(F(c'),F(c)) \xrightarrow{\varepsilon_d} \mathrm{Hom}_\mathcal D(F(c'),d)\) is an equivalence. Equivalently, this is the data of a representing object for the presheaf \(\mathcal C^\mathrm{op}\xrightarrow{\mathrm{Hom}_\mathcal D(F(-),d)} \mathcal S\) (which by definition comes equipped with the data of a universal element \(\varepsilon_d \in \mathrm{Hom}_\mathcal D(F(c),d)\) witnessing it as such). Then, a right adjoint exists if and only if a pointwise right adjoint exists at all objects of \(\mathcal D\); in this case, the right adjoint is the (necessarily unique) factorization of the functor \(\mathcal D\xrightarrow{\mathrm{Hom}_\mathcal D(F(=),-)} \mathcal P(\mathcal C)\) through the Yoneda embedding. (See Subsection A.5 for an alternative description of \(\infty\)-categorical adjunctions.)

As a basic example, there exists a right adjoint \(\mathcal S\xleftarrow{\iota_0} \mathrm{Cat}_\infty\) to the inclusion, which carries an \(\infty\)-category \(\mathcal C\) to its maximal subgroupoid \(\mathcal C^\simeq\) (which is obtained by discarding all of its noninvertible morphisms).