Coherence for weak units (Q1946063)

Summary: We define weak units in a semi-monoidal 2-category \(\mathbb C\) 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 \(\mathbb C\), and \(\alpha: I tensor I \to I\) is an equivalence in \(\mathbb C\). 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 {ref thmC}: the 2-category of weak units is contractible if non-empty. Finally we show: {Theorem E}. 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.