Calculating maps between n-categories

TL;DR

This paper presents a method for explicitly calculating homotopy classes of morphisms between Tamsamani n-categories using a novel notion of free cofibration and Reedy-cofibrant resolutions.

Contribution

It introduces a new concept of free cofibration for n-precats and applies it to compute homotopy classes of maps between n-categories.

Findings

Provides an explicit calculation method for homotopy classes of morphisms.

Introduces the notion of free cofibration for n-precats.

Establishes an analogy with Bousfield-Kan cofibrations.

Abstract

We give an explicit way of calculating the set of homotopy classes of morphisms from a Tamsamani n-category A to another one B. This calculation uses a Reedy-cofibrant cosimplicial resolution of A, using a new notion of ``free cofibration'' of n-precats. The free cofibrations of n-precats seem to be the analogue for n-categories of the Bousfield-Kan cofibrations in the theory of diagrams.