Coherence for weak units
2013; Volume: 18; Linguagem: Inglês
10.4171/dm/392
ISSN1431-0643
Autores Tópico(s)Advanced Operator Algebra Research
ResumoWe define weak units in a semi-monoidal 2-category \mathcal C 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 \mathcal C , and \alpha: I \otimes I \to I is an equivalence in \mathcal C . 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.
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