  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.
