nLab coherence theorem for bicategories with finite limits

Theorem

Theorem

(Power) Any bicategory with finite bilimits is equivalent to a strict 2-category with finite flexible limits.

Proof

Let $K$ be a bicategory with finite bilimits, let $K \hookrightarrow [K^{op},Cat]$ be its Yoneda embedding, and let $K'$ be the closure of $K$ in $[K^{op},Cat]$ under finite flexible limits. Since $Cat$ is a strict 2-category with finite flexible limits, so is $[K^{op},Cat]$. And since $K$ has finite bilimits, and these are preserved by its Yoneda embedding, while flexible limits are in particular bilimits, every object of $K'$ is equivalent to an object of $K$. Thus, $K\simeq K'$.

Furthermore, the 2-category of finite bilimit-preserving pseudofunctors into $Cat$ is equivalent to the 2-category of finite PIE limit-preserving 2-functors into $Cat$ and pseudonatural transformations.

References

• John Power, Coherence for bicategories with finite bilimits I, Categories in computer science and logic, Contemporary Mathematics 92 (1989) pp 341-347 MR1003207, doi:10.1090/conm/092, (Google Books)

• John Power, Three dimensional monad theory, Contemporary Mathematics 431 (2007): 405.