Homomorphisms of higher categories

TL;DR

This paper introduces a construction for homomorphisms in higher categories that preserve structure up to weak equivalence, with applications to tricategories and Batanin's omega-categories.

Contribution

It provides a new method to define homomorphisms in higher categories with associative composition, applicable to various models like tricategories and omega-categories.

Findings

Homomorphisms preserve structure up to weak invertibility.

Construction admits strictly associative and unital composition.

Applications to tricategories and Batanin's omega-categories.

Abstract

We describe a construction that to each algebraically specified notion of higher-dimensional category associates a notion of homomorphism which preserves the categorical structure only up to weakly invertible higher cells. The construction is such that these homomorphisms admit a strictly associative and unital composition. We give two applications of this construction. The first is to tricategories; and here we do not obtain the trihomomorphisms defined by Gordon, Power and Street, but only something equivalent in a suitable sense. The second is to Batanin's weak omega-categories.