  The notion of determinant groupoid is a natural outgrowth of the theory of
the Sato Grassmannian and thus well-known in mathematical physics. We briefly
sketch here a version of the theory of determinant groupoids over an artinian
local ring, taking pains to put the theory in a simple concrete form suited to
number-theoretical applications. We then use the theory to give a simple proof
of a reciprocity law for the Contou-Carr\`{e}re symbol. Finally, we explain how
from the latter to recover various classical explicit reciprocity laws on
nonsingular complete curves over an algebraically closed field, namely
sum-of-residues-equals-zero, Weil reciprocity, and an explicit reciprocity law
due to Witt. Needless to say, we have been much influenced by the work of Tate
on sum-of-residues-equals-zero and the work of Arbarello-DeConcini-Kac on Weil
reciprocity. We also build in an essential way on a previous work of the
second-named author.
