Graph-based reaction systems were recently introduced as a generalization of the intensely studied set-based reaction systems. They deal with simple edge-labeled directed graphs, and dynamic semantics of graph-based reaction systems is defined by graph surfing as a novel kind of graph transformation where, in a single surf step, reactions are applied to a subgraph of a given background graph yielding a successor subgraph. In this paper, we propose a categorical approach to reaction systems so that a wider spectrum of data structures becomes available on which reaction systems can be based. In this way, many types of graphs, hypergraphs, and graph-like structures are covered.