The concept of structured decompositions, as introduced in the context of graphs, can indeed be extended to other mathematical structures, including higher-dimensional topological spaces and algebraic varieties. This extension can be achieved by leveraging the categorical framework that underpins structured decompositions.
Higher-Dimensional Topological Spaces: In topology, one can consider simplicial complexes or CW complexes as higher-dimensional analogs of graphs. A structured decomposition in this context could involve associating categorical objects to the vertices, edges, and higher-dimensional faces of a simplicial complex. By defining a functor that maps these complexes to a suitable category, one can create structured decompositions that capture the combinatorial and topological properties of the space. For instance, one could use the notion of a "nerve" of a cover to construct a structured decomposition that reflects the connectivity and higher-dimensional relationships within the space.
Algebraic Varieties: In algebraic geometry, structured decompositions can be applied to algebraic varieties by associating geometric objects (such as schemes or sheaves) to the points and subvarieties of a given variety. The functorial approach allows for the definition of decompositions that respect the algebraic structure, such as morphisms between varieties. This could lead to new insights into the geometric properties of varieties, such as their dimension and singularities, by analyzing the structured decompositions in terms of their categorical representations.
General Framework: The key to extending structured decompositions lies in the ability to define a suitable category that captures the essential features of the mathematical structure in question. By identifying a "spine" functor that selects objects of interest within the category, one can generalize the notion of width and develop compositional algorithms that exploit the structural properties of these higher-dimensional objects.