nLab Lack's coherence observation

Idea

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.

Examples

Instances of Lack’s coherence observation include:

• The bicategory Prof of small categories and profunctors is equivalent to the strict 2-category of presheaf categories and cocontinuous functors.

• The bicategory Span(C) of spans in a category $C$ with pullbacks is equivalent to the strict 2-category of slice categories of $C$ and linear polynomial functors. More generally, the bicategory of polynomials in a locally cartesian closed category is equivalent to the strict 2-category of slices and polynomial functors.

Both examples are instances of a more general result. Given a relative pseudomonad $T$ on a strict 2-category $K$, the Eilenberg–Moore bicategory? of $T$ 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.

References

This observation first appears in Example 1.5(k) of

• Steve Lack, A 2-categories companion, in: John C. Baez and J. Peter May (Eds) Towards Higher Categories, The IMA Volumes in Mathematics and its Applications, vol 152, Springer, 2009, doi:10.1007/978-1-4419-1524-5_4, arXiv:math/0702535.

The observation that instances generally arise from Kleisli bicategories is in:

• Nathanael Arkor, Philip Saville, Andrew Slattery, Bicategories of algebras for relative pseudomonads, arXiv:2501.12510