Coherence for adjunctions in a $4$-category

TL;DR

This paper defines coherent adjunctions in 4-categories, proves certain restriction maps are trivial fibrations, and conjectures a generalization to n-categories, advancing the understanding of higher categorical adjunctions.

Contribution

It introduces a formal definition of coherent adjunctions in 4-categories and proves key properties about the restriction maps, with a conjecture extending to n-categories.

Findings

Restriction map from coherent adjunctions to 1-morphisms with adjoints is a trivial fibration.

Other restriction maps fixing parts of the adjunction data are also trivial fibrations.

Provides a conjectural framework for coherent adjunctions in n-categories.

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.