Coherence for weak units

André Joyal*, Joachim Kock

*Corresponding author for this work

Research output: Contribution to journalArticleResearchpeer-review

7 Citations (Scopus)


We define weak units in a semi-monoidal 2-category L 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 L, and α: I ⊗ I → I is an equivalence in L. 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.

Original languageEnglish
Pages (from-to)71-110
Number of pages40
JournalDocumenta Mathematica
Issue number1
Publication statusPublished - 2013


  • Coherence
  • Monoidal 2-categories
  • Units


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