  The notion of a Manin triple of Lie algebras admits a generalization, to dg
Lie algebras, in which various properties are required to hold only up to
homotopy.
  This paper introduces two classes of examples of such homotopy Manin triples.
These examples are associated to analogs in complex dimension two of,
respectively, the punctured formal 1-disc, and the complex plane with multiple
punctures. The dg Lie algebras which appear include certain higher current
algebras in the sense of Faonte, Hennion and Kapranov arXiv:1701.01368.
  We work in a ringed space we call rectilinear space, and one of the tools we
introduce is a model of the derived sections of its structure sheaf, whose
construction is in the spirit of the adelic complexes for schemes due to
Parshin and Beilinson.
