Coherence for adjunctions in a $4$-category

Abstract

We give a definition of a coherent adjunction in a $4$-category consisting of a finite list of $k$-morphisms for $k\leq 4$, plus equations beetween $4$-morphisms. We prove that the restriction map from the space of coherent adjunctions in a $4$-category to the space of $1$-morphisms which admit an adjoint is a trivial fibration. We prove that other restriction maps related to fixing parts of the data of an adjunction are also trivial fibrations. We give a conjectural description of a coherent adjunction in an $n$-category.