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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.