nLab coherence theorem for bicategories with finite limits


2-Category theory

Limits and colimits



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


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

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


  • 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.

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