# nLab Lack's coherence observation

Contents

### Context

#### 2-category theory

2-category theory

Definitions

Transfors between 2-categories

Morphisms in 2-categories

Structures in 2-categories

Limits in 2-categories

Structures on 2-categories

# Contents

## 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 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

Last revised on July 14, 2022 at 04:10:45. See the history of this page for a list of all contributions to it.