Symmetries as Isomorphisms

TL;DR

This paper proposes a categorical approach to treat symmetries as isomorphisms between models, applying it to electromagnetism and exploring implications for philosophy and foundations of mathematics.

Contribution

It introduces a rigorous categorical framework that formulates symmetries as isomorphisms, bridging models in physics and foundations in mathematics.

Findings

Categorical strategy successfully models symmetries as isomorphisms in electromagnetism.

Addresses ontological nonperspicuity using natural operators in algebraic models.

Highlights significance of symmetries as isomorphisms within Univalent Foundations.

Abstract

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.