COHERENCE FOR WEAK UNITS
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COHERENCE FOR WEAK UNITS. / Joyal, Andre; Kock, Joachim.
In: Documenta Mathematica, Vol. 18, 2013, p. 71-110.Research output: Contribution to journal › Journal article › Research › peer-review
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TY - JOUR
T1 - COHERENCE FOR WEAK UNITS
AU - Joyal, Andre
AU - Kock, Joachim
PY - 2013
Y1 - 2013
N2 - We define weak units in a semi-monoidal 2-category 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 C, and alpha : I circle times I -> I is an equivalence in C. We show that this notion of weak unit has coherence built in: Theorem A: a 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.
AB - We define weak units in a semi-monoidal 2-category 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 C, and alpha : I circle times I -> I is an equivalence in C. We show that this notion of weak unit has coherence built in: Theorem A: a 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.
KW - Monoidal 2-categories
KW - units
KW - coherence
M3 - Journal article
VL - 18
SP - 71
EP - 110
JO - Documenta Mathematica
JF - Documenta Mathematica
SN - 1431-0635
ER -
ID: 331501631