Coherence for weak units

TL;DR

This paper introduces a notion of weak units in semi-monoidal 2-categories, demonstrating their inherent coherence and equivalence to tricategory units, with implications for simplifying higher categorical structures.

Contribution

It defines weak units as cancellable pseudo-idempotents and proves their coherence properties, establishing their equivalence to units in tricategories.

Findings

Weak units have a canonical associator satisfying the pentagon equation.

Morphisms of weak units are automatically compatible with associators.

The 2-category of weak units is contractible if non-empty.

Abstract

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 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.