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# Coherence for weak units

By André Joyal and Joachim Kock
*Documenta Math.* **18** (2013), 71--110.
arXiv:0907.4553

#### Abstract

We define weak units in a semi-monoidal 2-category
*C* as cancellable pseudo-idempotents: they are pairs (*I*,α)
where *I* is an object such that tensoring with *I* from either side
constitutes a biequivalence of *C*, and α: *I* ⊗
*I* → *I*
is an equivalence in *C*. We show that this notion of weak unit has
coherence built in: Theorem A: α 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: α 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.
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Last updated: 2013-02-19 by
Joachim Kock.