First things ought to come first. And the first things in mathematics is arithmetic and Euclidean geometry. They were the first systems to be fully charactised first. Arithmetic, well before Euclid and Euclidean geometry by the eponymous Euclid. I say characterised rather than axiomatic as arithmetic was so simple that no-one bothered to axiomatise it until Peano in the early 20C. Now whilst these are thought as the iconic examples of mathematics, I would argue that they were the original physical theories that were the first ones to be axiomatised. We see examples of this being done today. For example, as one of the posts here mention, rational CFTs have been fully axiomatised.

Thus has category theory has anything to say about arithmetic? Well, traditionally, multiplication is seen as iterative addition. Category theory provides an alternative description of this as showing the coproduct (addition) is dual to the product as the naming suggests. This is new perspective on one of the oldest ideas in arithmetic is one reason why I  was attracted to category theory. But this notion doesn't stop there. The notion of a product and dually, the coproduct,  are available in any category as the universal property characterising these properties is categorical. Whether they exist is a seperate question. Thus we can ask whether they exist in categories of groups, rings, modules and algebras as well as other more esoteric categories. And they do and this gives a systematic way of proceeding compared to the adhoc, heuristic way they were first thought up. Now all these mathematical structures are important in physics: groups are important in physics as the correct way of thinking about symmetries and this is then naturally generalised to groupoids. Bundles are important in physics as a general description of field theories and their sections organise themselves into a module over the ring of functions one the base manifold. The infinitesimal description of a Lie group is a Lie algebra.

On a more sophisticated note, TQFTs are most elegantly presented via category theory. The original axioms were set out by Atiyah in the late 80s I believe.

First things ought to come first. And the first things in mathematics is arithmetic and Euclidean geometry. They were the first systems to be fully charactised first. Arithmetic, well before Euclid and Euclidean geometry by the eponymous Euclid. I say characterised rather than axiomatic as arithmetic was so simple that no-one bothered to axiomatise it until Peano in the early 20C. Now whilst these are thought as the iconic examples of mathematics, I would argue that they were the original physical theories that were the first ones to be axiomatised. We see examples of this being done today. For example, as one of the posts here mention, rational CFTs have been fully axiomatised.

Thus has category theory has anything to say about arithmetic? Well, traditionally, multiplication is seen as iterative addition. Category theory provides an alternative description of this as showing the coproduct (addition) is dual to the product as the naming suggests. This is new perspective on one of the oldest ideas in arithmetic is one reason why I  was attracted to category theory. But this notion doesn't stop there. The notion of a product and dually, the coproduct,  are available in any category as the universal property characterising these properties is categorical. Whether they exist is a seperate question. Thus we can ask whether they exist in categories of groups, rings, modules and algebras as well as other more esoteric categories. And they do and this gives a systematic way of proceeding compared to the adhoc, heuristic way they were first thought up. Now all these mathematical structures are important in physics: groups are important in physics as the correct way of thinking about symmetries and this is then naturally generalised to groupoids. Bundles are important in physics as a general description of field theories and their sections organise themselves into a module over the ring of functions one the base manifold. The infinitesimal description of a Lie group is a Lie algebra.

On a more sophisticated note, TQFTs are most elegantly presented via category theory. The original axioms were set out by Atiyah in the late 80s I believe. A field theory is topological when it has no local dynamics/dof and so the only dof are global.