Graph Neural Networks (GNNs) are almost universally built on a single primitive: the neighborhood. Regardless of architectural variations, message passing ultimately aggregates over neighborhoods, which intrinsically limits expressivity and often yields power no stronger than the Weisfeiler-Lehman (WL) test. In this work, we challenge this primitive. We introduce the Grothendieck Graph Neural Networks (GkGNN) framework, which provides a strict algebraic extension of neighborhoods to covers, and in doing so replaces neighborhoods as the fundamental objects of message passing. Neighborhoods and adjacency matrices are recovered as special cases, while covers enable a principled and flexible foundation for defining topology-aware propagation schemes. GkGNN formalizes covers and systematically translates them into matrices, analogously to how adjacency matrices encode neighborhoods, enabling both theoretical analysis and practical implementation. Within this framework, we introduce the cover of sieves, inspired by category theory, which captures rich topological structure. Based on this cover, we design Sieve Neural Networks (SNN), a canonical fixed-cover instantiation that generalizes the adjacency matrix. Experiments show that SNN achieves zero observed failures on challenging graph isomorphism benchmarks (SRG, CSL, BREC) and substantially improves topology-aware evaluation via a controlled label-propagation probe. These results establish GkGNN as a principled foundational framework for replacing neighborhoods in GNNs.