Categorification of the Periodic system of mathematical models: the 2-category PSM2 and dualities as higher morphisms (Part V)

Building upon the multidimensional classification of the Periodic System of Mathematical Models (PSMM) developed in previous parts, we transition from a static description to a categorical one: a 2-category, denoted PSM2, is introduced. Its objects are the cells of the PSMM, 1-morphisms are transitions between cells (changes in coordinates), and 2-morphisms are equivalences between different paths. All 1-morphisms known from previous articles (the matrix bridge, complexification, quantization, and limiting transitions) are systematized. Concrete examples demonstrate the emergence of 2-morphisms: the commutativity of complexification and projectivization, the relationship between two methods of constructing a hyperelliptic curve, the connection between quantization and the classical limit, and mirror symmetry for elliptic curves. It is shown that fundamental dualities of modern mathematics and physics - mirror symmetry, the AdS/CFT correspondence, S-duality, and the geometric Langlands program - are naturally interpreted as 2-morphisms in PSM2. An algorithm for the systematic search for new dualities, based on the analysis of commutative diagrams, is developed; this algorithm is used to formulate a series of hypotheses connecting the cells of hyperelliptic curves, spectral graphs, integrable systems, random matrices, and tropical models. Finally, a path towards an -categorification of the PSMM is outlined, and the concept of a stack of models is introduced, opening a perspective for unifying category theory, algebraic geometry, and mathematical physics into a single framework.