Coherence for Weak Units

We define weak units in a semi-monoidal $2$-category $\CC$ 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 $\CC$, and $\alpha: I \tensor I \to I$ is an equivalence in $\CC$. We show that this notion of weak unit has coherence built in: Theorem \ref{thmA}: $\alpha$ has a canonical associator $2$-cell, which automatically satisfies the pentagon equation. Theorem \ref{thmB}: 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 \ref{thmE}) 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.

2010 Mathematics Subject Classification: 18D05; 18D10

Keywords and Phrases: Monoidal $2$-categories, units, coherence.

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