Symmetries as Isomorphisms

Symmetries and isomorphisms play similar conceptual roles when we consider how models represent physical situations, but they are formally distinct, as two models related by symmetries are not typically isomorphic. I offer a rigorous categorical strategy that formulate symmetries as isomorphisms between models and apply it to classical electromagnetism, and evaluate its philosophical significance in relation to the recent debate between `sophistication' and `reduction'. In addition to traditional spacetime models, I also consider algebraic models, in which case we can use the method of natural operators to address the problem of ontological nonperspicuity faced by the categorical strategy. Finally, I briefly expound on the significance of symmetries as isomorphisms in the framework of Univalent Foundations, in which isomorphic structures are formally identified.