  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.
