Strictifying Operational Coherences and Weak Functor Classifiers in Low Dimensions

Abstract

Weak structures abound in higher category theory, but are often suitably equivalent to stricter structures that are easier to understand. We extend strictification for tricategories and trihomomorphisms to trinatural transformations, trimodifications and perturbations. Along the way we distinguish between the operational coherences, which are possible to strictify, and the coherences on globular inputs, which remain weak. We introduce generalised path objects for $\mathbf{Gray}$-categories, which help reduce proofs in the three-dimensional setting to known results. Upon closing the resulting semi-strict trinatural transformations under composition, we state the hom-triequivalences of what we expect to be a `semi-strictification tetra-adjunction'.