Categorical central extensions and reciprocity laws

The local object, constructed on a two-dimensional arithmetical scheme, is
a two-dimensional local field. Such a field is a complete discrete
valuation field with a residue field again a complete discrete valuation
field with finite residue field. Two-dimensional local fields naturally
appear from points and formal stalks of arithmetical curves on
two-dimensional arithmetical schemes.  The two-dimensional class field
theory was developed in 70-s and 80-s by K.Kato, A.N. Parshin and others
in terms of Milnor K-groups of two-dimensional local fields. The
reciprocity laws connect the two-dimensional local and global class field
theories. I will explain how the reciprocity laws for the two-dimensional
tame symbol on an algebraic surface can be proved by using higher
categories and two-dimensional adeles. I will explain also how this method
gives a lot of new reciprocity laws on a surface when we change the ground
field to an Artinian ring. The talk is based on joint results with
Xinwen Zhu.