André Joyal and Joachim Kock

摘要

We define weak units in a semi-monoidal $2$-category $CC$ as cancellable pseudo-idempotents: they are pairs $(I,lpha)$ where $I$ is an object such that tensoring with $I$ from either side constitutes a biequivalence of $CC$, and $lpha: I ensor I o I$ is an equivalence in $CC$. We show that this notion of weak unit has coherence built in: Theoremef{thmA}: $lpha$ has a canonical associator $2$-cell, which automatically satisfies the pentagon equation. Theoremef{thmB}: every morphism of weak units is automatically compatible with those associators. Theoremef{thmC}: the $2$-category of weak units is contractible if non-empty. Finally we show (Theoremef{thmE}) that the notion of weak unit is equivalent to the notion obtained from the definition of tricategory: $lpha$ 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.