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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJun 8th 2025
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexDeleted. Former content: This page contains ideas of concepts and constructions which might be profitably [[vertical categorification|categorified]]. ###Galois theory for algebraic extensions### Jeff Morton and John Baez <a href="http://www.physicsbanter.com/742075-post12.html">apparently</a> >... 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](http://golem.ph.utexas.edu/category/2008/04/2structure_types.html). ###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](http://golem.ph.utexas.edu/category/2007/08/justificatory_narratives.html#c011603). <a href="https://ncatlab.org/nlab/revision/diff/things+to+be+categorified+%3E+history/8">diff</a>, <a href="https://ncatlab.org/nlab/revision/things+to+be+categorified+%3E+history/8">v8</a>, <a href="https://ncatlab.org/nlab/show/things+to+be+categorified+%3E+history">current</a>

   Deleted. Former content:

   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.

   diff, v8, current