Adjointable functor

Definition 15.1.1. Let \((C,C,C_R)\) be an adequate triple and let \(F\colon C\catop \to \Cat _{\infty }\) be a functor.

(1)

The functor \(F\) is left \(C_R\)-adjointable if, for every \(r\colon X\to Y\) in \(C_R\), the functor \(r^*:=F(r)\colon F(Y)\to F(X)\) admits a left adjoint \(r_!\), and for every pullback square

with \(r,r'\in C_R\), the Beck–Chevalley transformation \[ r'_!{l'}^* \xrightarrow {r'_!{l'}^*\eta _{r}} r'_!{l'}^*r^*r_! \simeq r'_!{r'}^*l^*r_! \xrightarrow {\epsilon _{r'}l^*r_!} l^*r_! \] is a natural isomorphism.

(2)

The functor \(F\) is right \(C_R\)-adjointable if every \(r^*\) admits a right adjoint \(r_*\) and, for every pullback square as above, the dually defined Beck–Chevalley transformation \(l^*r_*\to r'_*{l'}^*\) is a natural isomorphism.

When \(C_R=C\), we simply say that \(F\) is left adjointable or right adjointable.