Pseudofunctor

Summary

A pseudofunctor is a map between bicategories that preserves composition and identities up to specified invertible 2-morphisms, rather than on the nose. These comparison 2-morphisms must be compatible with the associators and unitors. Pseudofunctors into the 2-category of categories describe families of categories that vary with a parameter. Examples are the categories of modules over varying rings, or of sheaves over varying spaces. The Grothendieck construction turns such a pseudofunctor into a fibred category.