… 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.

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.

Revised on July 4, 2009 11:19:14
by Toby Bartels
(71.104.230.172)