This page contains ideas of concepts and constructions which might be profitably categorified.
Jeff Morton and John Baez apparently
… figured out how to categorify the algebraic integers in any algebraic extension of the rationals, getting an “algebraic extension” of the category of finite sets. We figured out the beginnings of a theory that associates a “Galois 2-group” to any such algebraic extension.
The Yoneda embedding of a category to is a 2-coalgebra for the 2-endofunctor .
Like with the powerset functor there surely can’t be a terminal 2-coalgebra for . What about , where is a fixed category? The terminal 2-coalgebra would have as objects finite or infinite lists of objects of with lists of arrows of as morphisms.
A sketch of 2-structure types.
For any representation of the fundamental group of a punctured Riemann surface, we can find a linear differential equation with holomorphic coefficients, such that the monodromies of the solutions realize this representation: discussion.