nLab Lack's coherence observation




One form of the coherence theorem for bicategories states that every bicategory is equivalent to a strict 2-category. Lack’s coherence observation (named for Steve Lack) states that naturally occurring bicategories tend to be equivalent to naturally occurring strict 2-categories.


Instances of Lack’s coherence observation include:

The first example is an instance of a more general result. Given a relative pseudomonad TT on a strict 2-category KK, the Eilenberg–Moore bicategory? of TT will also be a strict 2-category, whereas the Kleisli bicategory will usually not be. However, the Kleisli bicategory embeds fully faithfully into the Eilenberg–Moore bicategory, and so will be biequivalent to the full sub-2-category of the Eilenberg–Moore bicategory on the free algebras.


This observation first appears in Example 1.5(k) of

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