Coherence for weak units
André JoyalDépartement de Mathématiques Departament de Matemàtiques Université du Québec à Montréal Univ. Autònoma de Barcelona
Joachim KockDépartement de Mathématiques Departament de Matemàtiques Université du Québec à Montréal Univ. Autònoma de Barcelona
We define weak units in a semi-monoidal 2-category as cancellable pseudo-idempotents: they are pairs where is an object such that tensoring with from either side constitutes a biequivalence of , and is an equivalence in . We show that this notion of weak unit has coherence built in: Theorem refthmA: has a canonical associator 2-cell, which automatically satisfies the pentagon equation. Theorem refthmB: every morphism of weak units is automatically compatible with those associators. Theorem refthmC: the 2-category of weak units is contractible if non-empty. Finally we show (Theorem refthmE) that the notion of weak unit is equivalent to the notion obtained from the definition of tricategory: 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.
Cite this article
André Joyal, Joachim Kock, Coherence for weak units. Doc. Math. 18 (2013), pp. 71–110DOI 10.4171/DM/392