**9. Conclusions**

We provided a general theoretical and conceptual investigation of nonlocal continuum field theories that aimed to achieve several goals. First, the subject was revisited from a new perspective, with the intention of introducing it not only to mathematical and theoretical physicists, but also invite a wider audience, including engineers, material scientists, chemists, applied physicists, and applied mathematicians. The various essential

ideas behind nonlocality in material continua were put under new light with the help of an abstract field-response model developed in three dimensions. Next, the fine-grained topological microstructure of nonlocal metamaterials was explicated in detail. We introduced the concept of physics-based nonlocality microdomains, demonstrating how the latter regions present an important structural topological feature of the physics of nonlocal media. Afterwards, it was proved using differential topology that a natural fiber bundle structure, serving as a "source (excitation) superspace", can be constructed. The material source fiber bundle superspace, or the material superspace in short, was shown to possess all of the required properties of a standard fiber bundle ye<sup>t</sup> while faithfully reflecting the physics of nonlocal microdomains. Eventually, and using the technique of partition of unity, it was proved that the fiber bundle superspace can be deployed for the purpose of constructing and computing the complete nonlocal material response function over arbitrary microdomain configurations. This was accomplished by building a bundle homomorphism to replace the well-known, but now inadequate, material tensor linear operators commonly utilized in local continuum field theories, for instance, conventional electromagnetism. This new homomorphism can be viewed as a generalization of the linear operators of the various classical boundary-value problems of mathematical physics. It is hoped that in the future this homomorphism may be "topologically discretized" using suitable methods borrowed from other advanced fields such as algebraic topology, computational topology, and global analysis. The new fiber bundle superspace formulation suggested that nonlocal continuum field theories could be reformulated in an alternative way compared with the prevailing existing methods. Most importantly, nonlocality in material continua forces us to introduce an entire array of infinite-dimensional Banach spaces attached to every point in the conventional three-dimensional base space inside which the material is conventionally defined. This extra or additional fiber structure provides a natural explanation of why traditional boundary conditions often fail to account for the physics of nonlocal metamaterials. Moreover, the fiber bundle theory opens the door for several new applications, including the ability to understand the deep connection between topology and field theories, e.g., electromagnetism, in engineered artificial media. Overall, the author proposes that future research in metamaterials will gradually require more extensive collaboration between engineers and mathematicians in order to explore in full the deep consequences of this organic topology/electromagnetism relation.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The author declares no conflict of interest.
