This dissertation explores new directions in the application of enriched category theory to computational problems.We pay special attention to the use of quantale-enriched categories, particularly studying the representation theory of complete quantale-enriched categories (weighted lattices) and networked diffusion between such categories. Our representation theory is rooted in formal concept analysis, an application of lattice theory to data analysis. We generalize a categorical treatment of Mori to the quantale-enriched setting. Concept analysis establishes an equivalence between the category of so-called formal contexts and the category of sup-lattices. This equivalence provides practical computational methods for working with weighted lattices. Building on this foundation, we study networked diffusion processes valued in weighted lattices. We introduce the Lawvere Laplacian, an operator analogous to the sheaf Laplacian of Hansen and Ghrist \cite{hansen2019}. This operator synthesizes local data through weighted limits, analogous to linear combinations in vector spaces, and reveals that many results from the linear algebraic setting have analogs for weighted lattices. Specifically, we prove that the category of suffix points of the Lawvere Laplacian forms a complete $\Q$-category and establish an analogy to the Hodge theorem in differential topology. This framework accommodates the transfer of data susceptible to error through fuzzy adjoint functors, where the resulting discrete-time heat equation converges to a fuzzy global section up to a quantale-valued error. This approach finds applications in diverse domains, including discrete event systems, network optimization problems, and preference diffusion models. Finally, we develop a categorical framework for studying a generalized form of conic programming. We generalize the notion of a linear cone in conic programming, obtaining a definition applicable in an arbitrary category. This formulation accommodates both classical cones in vector spaces and novel cone-like structures in categories such as weighted lattices and relations. We provide generalized proofs of many foundational results in conic programming, such as the correspondence between pointed convex cones and linear preorders. Using the Chu construction, we build a duality theory for generalized conic programs and ultimately prove a weak duality theorem valid in a suitably structured category. Throughout this thesis, we demonstrate that many fundamental properties of computational processes depend primarily on underlying order-theoretic and compositional structure, rather than on specific metric or linear-algebraic properties. We hope that exploring this alternative perspective may reveal new connections between seemingly disparate problems and suggest fresh directions for both theoretical developments and practical applications.