Weak identity arrows in higher categories

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 Δ 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, including Moore path spaces and cobordism categories. The category of fair 2-categories is shown to be equivalent to the category of bicategories with strict composition laws. 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.