# nLab Notes on geometric Langlands

This entry is about the collection of notes

on the categorical geometric Langlands conjecture, which is formulated as an equivalence of stable (infinity,1)-categories of

and

where $X$ is a smooth complete curve over a field of characteristic zero, and $G$ is a reductive group and $G^\vee$ is its Langlands dual?. (More precisely, the latter category should be replaced by some appropriate category of ind-coherent sheaves.)

• D. Gaitsgory et. al., Geometric representation theory, graduate seminar, Fall 2009–Spring 2010, web.

## Contents

### Foundational stuff

Studies some aspects of the symmetric monoidal (infinity,1)-category of dg-categories, including colimits, limits, and dualizable objects in it, and categories of modules over it.

An explicit description of filtered colimits in the (infinity,1)-category of (infinity,1)-categories.

A concise review of derived stacks and derived schemes in derived algebraic geometry (the version based on coconnective commutative dg-algebras).

A concise review of stable (infinity,1)-categories of quasi-coherent sheaves and perfect complexes on derived stacks.

On the stable (infinity,1)-category of ind-coherent sheaves on a derived stack, and the six operations in this setting.

Develops the theory of ind-schemes in derived algebraic geometry.

Studies crystals and D-modules in derived algebraic geometry.

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rest of the contents to be filled in

Created on January 25, 2015 at 13:23:00. See the history of this page for a list of all contributions to it.