1.2. Morphisms between locales. We introduced the above ideas using a map of simplicial sets f : X → Y , but in actuality the definitions of F∗ and F∗ work whenever F is a open morphism of locales, as we shall soon see. To get to that point, however, it is best to start all over, by taking X and Y to be arbitrary locales and considering an arbitrary adjunction f∗ : Y //X : f∗ oo as a morphism F...

We show that an adjoint functor between quasi-categories may be extended to a simplicially enriched functor whose domain is an explicitly presented “homotopy coherent adjunction”. This enriched functor encapsulates both the coherent monad and the coherent comonad generated by the adjunction. Furthermore, because its domain is cofibrant, this data can be used to construct explicit quasi-categori...