TITLE: 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.  I would like to present a
new approach where instead the notion of identity arrow is weakened ---
these are tentatively called fair categories.  The approach is simplicial in
spirit: while strict higher categories can be described inductively as
certain Delta\op-diagrams with a strict Segal condition, the weakening of
the identity axiom is obtained by using a 'category of coloured ordinals'
instead of the usual Delta, and the diagrams are required to send coloured
ordinal maps to equivalences.  Every fair 1-category is isomorphic to a
plain category, and every fair 2-category is equivalent to a strict
2-category.  As applications: the simplicial localisation of a monoidal
model category is shown to be a fair monoidal simplicial category (i.e. a
fair simplicial 2-category with only one object), even if the unit is not
cofibrant.  Second application: there is a fair topological category of
Moore paths in a topological space.  If time permits I will also talk a bit
about application to extended TQFTs, and about a conjecture of Simpson on
realisation of homotopy n-types.