Why does coherence begin to matter at the tricategorical level?

It is well known that every weak $2$-category is equivalent to a strict $2$-category, with the equivalence essentially given by (unless I'm mistaken) the $2$-Yoneda embedding for an arbitrary $2$-category into its strict $2$-category of $2$-presheaves into $\mathfrak{Cat}$ -- this means we can effectively ignore the coherence diagrams for associativity and unitarity in $2$-categories and work with strict ones where composition is associative on the nose and units vanish under composition.

This is not so for $3$-categories; Gordon, Power and Street proved that all weak $3$-categories are equivalent to Gray categories, where associativity and unitarity still hold on the nose, but we have an additional coherence notion in play called 'interchange' for $2$-morphisms which introduces fundamental differences between $3$-categories with strict interchange and weak interchange laws.

Intuitively speaking, why does this type of coherence 'matter' more than the associative and unital coherence that appears at the $2$-categorical level?

The notions of '$n$-equivalence' become more varied as we move into higher $n$ as I understand it, so perhaps the 'correct' notion of $3$-equivalence is sensitive to interchange in some way that $2$-equivalence isn't sensitive to associativity for $1$-cells?

I am currently working through the linked Gordon-Power-Street paper on coherence for tricategories and suspect the answer is buried in their proof of the tricategorical coherence theorem, but I thought someone here might already be familiar with it and able to help a newcomer -- any assistance is appreciated.

  • 1
    $\begingroup$ You may want to look at Nick Gurski's fully algebraic treatment of coherence in tricategories ``Coherence in three-dimensional category theory''. $\endgroup$
    – Peter May
    Jul 8, 2019 at 14:37
  • $\begingroup$ @PeterMayThat looks very relevant, muchas gracias. $\endgroup$
    – Alec Rhea
    Jul 8, 2019 at 14:59

1 Answer 1


The short answer is that what matters is the combination of strict interchange and strict units, because interchange and units are what go into the Eckmann-Hilton argument, and dimension 3 is the first dimension in which you can have weak commutativity (e.g. braided monoidal categories are degenerate tricategories). GPS choose to strictify the units and therefore not the interchange, but you can also choose to strictify the interchange and not the units:

André Joyal, Joachim Kock, Weak units and homotopy 3-types, in: Street Festschrift: Categories in algebra, geometry and mathematical physics, Contemp. Math 431 (2007), 257-276, doi:10.1090/conm/431/08277, arXiv:math/0602084.

  • $\begingroup$ Much appreciated! $\endgroup$
    – Alec Rhea
    Jul 8, 2019 at 6:59

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