Geodesic
Coherence for weak units
Documenta mathematica, Tome 18 (2013), pp. 71-110.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: We define weak units in a semi-monoidal 2-category $\CC$ 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 $\CC$, and $\alpha: I \tensor I \to I$ is an equivalence in $\CC$. We show that this notion of weak unit has coherence built in: Theorem refthmA: $\alpha$ 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: $\alpha$ 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.
Classification : 18D05, 18D10
Keywords: monoidal 2-categories, units, coherence 
@article{DOCMA_2013__18__a46,
     author = {Joyal, Andr\'e and Kock, Joachim},
     title = {Coherence for weak units},
     journal = {Documenta mathematica},
     pages = {71--110},
     publisher = {mathdoc},
     volume = {18},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2013__18__a46/}
}
TY  - JOUR
AU  - Joyal, André
AU  - Kock, Joachim
TI  - Coherence for weak units
JO  - Documenta mathematica
PY  - 2013
SP  - 71
EP  - 110
VL  - 18
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DOCMA_2013__18__a46/
LA  - en
ID  - DOCMA_2013__18__a46
ER  - 
%0 Journal Article
%A Joyal, André
%A Kock, Joachim
%T Coherence for weak units
%J Documenta mathematica
%D 2013
%P 71-110
%V 18
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DOCMA_2013__18__a46/
%G en
%F DOCMA_2013__18__a46
Joyal, André; Kock, Joachim. Coherence for weak units. Documenta mathematica, Tome 18 (2013), pp. 71-110. http://geodesic.mathdoc.fr/item/DOCMA_2013__18__a46/