Functorial Renormalization: A Categorical Framework for Scale Transformations

We propose a categorical formulation of renormalization in which scale changes are en
coded as functors between categories (or topoi) that represent physical models at distinct res
olution scales. Renormalization is presented as a coherent family of functors Rs→t between
 scale-categories Cs and Ct (for ultraviolet s towards infrared t), equipped with natural trans
formations encoding coarse-graining maps and the algebra of observables’ projection. Under
 mild completeness hypotheses we conjecture the existence of universal fixed objects — cate
gorical RG fixed points — characterized by limit/colimit universality. Diagrammatic examples
 illustrate the finite-lattice → continuum passage. Consequences for photonic RG flow and con
nections to recent photonics/G-Theory studies are indicated and testable toy-model checks are
 suggested. This work aims to bridge categorical methods and renormalization practice, with
 particular eye toward photonic theories and De Ceuster’s G-Theory diagnostics.