The (Ω,ω)-width approach provides a powerful framework for defining width measures in a categorical context, but it does have certain limitations:
Dependence on Adhesiveness: The requirement that the category is adhesive can restrict the applicability of the (Ω,ω)-width approach. While many combinatorial objects reside in adhesive categories, there are important classes of structures that do not meet this criterion. This limitation may hinder the development of compositional algorithms for a broader range of mathematical objects.
Complexity of Width Measures: The definition of width as a category rather than a numeric invariant can complicate the interpretation and comparison of different width measures. This abstraction may obscure the practical implications of width in algorithmic contexts, making it challenging to derive concrete algorithmic results.
Alternative Definitions: To address these limitations, alternative approaches to defining width measures could be explored. For instance, one could consider numeric invariants that capture structural complexity while still allowing for categorical interpretations. This could involve defining width in terms of specific properties of morphisms or subobjects, potentially leading to more intuitive and computable measures.
General Compositional Algorithms: By exploring alternative definitions of width that do not rely on the adhesive property, one could develop more general compositional algorithms applicable to a wider variety of mathematical structures. This could involve integrating insights from other areas of mathematics, such as homotopy theory or category theory, to create a richer framework for understanding compositionality across different contexts.