Category theory offers a rich set of abstractions for the unified representation and manipulation of combinatorial objects such as relational databases, graphical scientific models, neural network architectures, and more! This project focused on leveraging these abstractions to define generic methods for creating and searching spaces of such objects. This necessitated robust implementations of categorical concepts such as limits and colimits in diagram categories as well as rewriting systems based on double pushout rewriting. As a technical contribution, our categorical rewriting system greatly outperformed state-of-the-art competitors at typed graph rewriting in terms of both memory usage and speed. On a practical note, we demonstrated how our software can create and search epidemiologically relevant subspaces of disease dynamic models.