On formalization of bicategory theory

Abstract

Bicategory theory is important in dealing with category theory from the categorical point of view. 2-categories and monoidal categories are special instances of bicategories. Although the Yoneda lemma for bicategories was expected to hold, it had not been rigorously proved before because its proof involves highly complex algebraic structures. The bicategorical Yoneda embedding is the most important corollary of the above lemma and it had not been rigorously proved, either. In this paper, we formalize the bicategorical Yoneda embedding directly on Extended Calculus of Constructions and report its implementation under the proof-checker LEGO. We then point out problems on our implementation and examine required functions of a proof-checker for the formalization of bicategory theory. The formalization signifies that application of a proof-checker is so effective to prove theorems which involve complex algebraic structures such as bicategories.KeywordsType TheoryNatural TransformationFormal ProofCategory TheoryNatural IsomorphismThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.