Weak identity arrows in higher categories

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 category Δ of "coloured ordinals," where the degeneracy maps are only up to homotopy. The main motivation for the theory is Simpson's weak-unit conjecture according to which η-groupoids with strict composition laws and weak units should model all homotopy η-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.