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:

Both examples are instances 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.

It is not clear whether there are other examples that are not special cases of the observation above.


This observation first appears in Example 1.5(k) of

Last revised on June 2, 2023 at 11:52:13. See the history of this page for a list of all contributions to it.