Coherence for weak units

Abstract

We define weak units in a semi-monoidal 2-category $\mathcal{C}$ 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 $\mathcal{C}$, and $\alpha :I\otimes I\to I$ is an equivalence in $\mathcal{C}$. We show that this notion of weak unit has coherence built in: Theorem A: $\alpha$ 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.