nLab things to be categorified

This page contains ideas of concepts and constructions which might be profitably categorified.

Galois theory for algebraic extensions

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.

Coalgebra

The Yoneda embedding of a category $\mathcal{C}$ to $F(\mathcal{C}) = [\mathcal{C}^{op}, Set]$ is a 2-coalgebra for the 2-endofunctor $F$.

Like with the powerset functor there surely can’t be a terminal 2-coalgebra for $F$. What about $G(\mathcal{C}) = 1 + A \times \mathcal{C}$, where $A$ is a fixed category? The terminal 2-coalgebra would have as objects finite or infinite lists of objects of $A$ with lists of arrows of $A$ as morphisms.

Structure Types

A sketch of 2-structure types.

Riemann-Hilbert problem

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.