Weak identity arrows in higher categories

TL;DR

This paper introduces fair categories, a new approach to weak higher categories that weakens identity arrows instead of composition, using a simplicial framework with homotopy-degenerate degeneracy maps.

Contribution

It proposes fair categories as a novel framework for higher categories, establishing equivalences with bicategories and tricategories with strict composition laws.

Findings

Fair 2-categories are equivalent to bicategories with strict composition.

Fair 3-categories correspond to tricategories with strict composition.

A version of Simpson's weak-unit conjecture is proved in dimension 3.

Abstract

There are a dozen definitions of weak higher categories, all of which loosen the notion of composition of arrows. A new approach is presented here, where instead the notion of identity arrow is weakened -- these are tentatively called fair categories. The approach is simplicial in spirit, but the usual simplicial category $\Delta$ is replaced by a certain `fat' delta of `coloured ordinals', where the degeneracy maps are only up to homotopy. The first part of this exposition is aimed at a broad mathematical readership and contains also a brief introduction to simplicial viewpoints on higher categories in general. It is explained how the definition of fair $n$-category is almost forced upon us by three standard ideas. The second part states some basic results about fair categories, and give examples. The category of fair 2-categories is shown to be equivalent to the category of bicategories with strict composition law. Fair 3-categories correspond to tricategories with strict composition laws. The main motivation for the theory is Simpson's weak-unit conjecture according to which $n$-groupoids with strict composition laws and weak units should model all homotopy $n$-types. A proof of a version of this conjecture in dimension 3 is announced, obtained in joint work with A. Joyal. Technical details and a fuller treatment of the applications will appear elsewhere.