This paper generalizes the notion of local coefficients and fundamental groups of spaces to simplicial coalgebras. We define a Hopf algebra
π1(C) from a simplicial coalgebra
C as a generalization of a fundamental group, and show that a module over
π1(C) corresponds to a local coefficient of
C. As a consequence, the Hoschild cohomology of a Hopf algebra
H with a coefficient
M coincides with the cohomology of the nerve simplicial coalgebra of
H with the local coefficient
M∗ associated with
M.
View full abstract