We introduce structured decompositions: category-theoretic generalizations of
many combinatorial invariants -- including tree-width, layered tree-width,
co-tree-width and graph decomposition width -- which have played a central role
in the study of structural and algorithmic compositionality in both graph
theory and parameterized complexity. Structured decompositions allow us to
generalize combinatorial invariants to new settings (for example decompositions
of matroids) in which they describe algorithmically useful structural
compositionality. As an application of our theory we prove an algorithmic meta
theorem for the Sub_P-composition problem which, when instantiated in the
category of graphs, yields compositional algorithms for NP-hard problems such
as: Maximum Bipartite Subgraph, Maximum Planar Subgraph and Longest Path