Yes, the connections between structured decompositions and other category-theoretic notions such as spined categories, undirected wiring diagrams, and monoidal width present rich avenues for exploration and exploitation:
Spined Categories: The relationship between structured decompositions and spined categories can be further investigated by examining how the spine functor can be generalized or adapted to different contexts. By understanding the interplay between the spine and the structured decomposition, researchers can potentially uncover new insights into the nature of compositionality and structural complexity in various mathematical settings.
Undirected Wiring Diagrams: The similarity between structured decompositions and undirected wiring diagrams suggests that there may be a deeper categorical framework that unifies these concepts. Exploring this connection could lead to the development of new tools and techniques for analyzing complex systems, particularly in areas such as network theory and systems biology, where wiring diagrams are prevalent.
Monoidal Width: The notion of monoidal width, which measures the complexity of morphisms in a monoidal category, can be related to structured decompositions by considering how the decomposition of objects can inform the understanding of morphisms. By establishing a bridge between these two concepts, one could develop new width measures that capture both object and morphism complexity, leading to more comprehensive compositional algorithms.
Interdisciplinary Applications: The exploration of these connections can also have interdisciplinary implications, as the ideas of structured decompositions and their related concepts can be applied to fields such as computer science, physics, and biology. By leveraging the categorical framework, researchers can develop new models and algorithms that reflect the underlying structural properties of complex systems across various domains.
In summary, the interplay between structured decompositions and other category-theoretic notions offers a fertile ground for further research, potentially leading to new insights and advancements in both theoretical and applied mathematics.