I wonder whether there is a definition of symmetry in mathematics that were not restricted to groups of symmetry and geometry. More precisely, a unified definition that makes sense in any branch of mathematics: Analysis, Topology, Algebra, Logic, etc. I would appreciate examples of symmetry in these areas of mathematics.
An example: In the x-y plane we see that the definition of symmetry of a set of points on a straight line. In this case, symmetry are functions (which form a symmetry group) that alter the shape of the figure. These functions define the so-called symmetrical relationship.
I suspect that this response is in Category Theory. But I'm not sure.