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.

The first example is an instance 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.

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.