The categorical geometric Langlands conjecture is a categorical version of the geometric Langlands conjecture. It 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.
Gurbir Dhillon, An informal introduction to categorical representation theory and the local geometric Langlands program (arXiv:2205.14578)
D. Gaitsgory, N. Rozenblyum, A study in derived algebraic geometry.
D. Gaitsgory et. al., Geometric representation theory, graduate seminar, Fall 2009–Spring 2010, web.
MO/56571, A precise statement of the categorical version of geometric Langlands conjecture.
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