COHERENCE FOR WEAK UNITS

We define weak units in a semi-monoidal 2-category C as cancellable pseudo-idempotents: they are pairs (I, alpha) where I is an object such that tensoring with I from either side constitutes a biequivalence of C, and alpha : I circle times I -> I is an equivalence in C. We show that this notion of weak unit has coherence built in: Theorem A: a has a canonical associator 2-cell, which automatically satisfies the pentagon equation. Theorem B: every morphism of weak units is automatically compatible with those associators. Theorem C: the 2-category of weak units is contractible if non-empty. Finally we show (Theorem E) that the notion of weak unit is equivalent to the notion obtained from the definition of tricategory: alpha alone induces the whole family of left and right maps (indexed by the objects), as well as the whole family of Kelly 2-cells (one for each pair of objects), satisfying the relevant coherence axioms.