# Tricategorical coherence

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.

• You may want to look at Nick Gurski's fully algebraic treatment of coherence in tricategories Coherence in three-dimensional category theory''. Jul 8, 2019 at 14:37
• @PeterMayThat looks very relevant, muchas gracias. Jul 8, 2019 at 14:59