  In the realm of invertible symmetry, the topological approach based on
classifying spaces dominates the classification of 't Hooft anomalies and
symmetry protected topological phases. We explore the alternative algebraic
approach based on cochains that directly characterize the lattice lagrangian of
invertible field theories and the anomalous phase factor of topological
operator rearrangements. In the current literature, the algebraic approach has
been systematically described for only finite 0-form symmetries. In this
initial work, we generalize it to finite higher-form symmetries with trivial
higher-group structure. We carefully analyze the algebraic cochains and
abstract a purely algebraic structure that naturally generalizes group
cohomology. Using techniques from simplicial homotopy theory, we show its
isomorphism to the cohomology of classifying spaces. The proof is based on an
explicit construction of Eilenberg-MacLane spaces and their products.
