nLab
coherence theorem for bicategories with finite limits

Context

2-Category theory

Limits and colimits

Theorem

Theorem

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

Proof

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'.

References

  • John Power, “Coherence for bicategories with finite bilimits”, Categories in computer science and logic, 1989.

Revised on October 6, 2012 14:31:36 by Urs Schreiber (82.113.99.144)