# A general definition of symmetry in mathematics

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.

Thank you.

• This "timeline" of the evolution of the idea of symmetry in different sciences and mathematics might help give some perspective for your question: theophys.kth.se/mathphys/SYM/sym_history.html – Joseph Malkevitch Dec 29 '11 at 14:34
• TIP If you wish to obtain further answers - perhaps some much more relevant and/or understandable - then it is never a good idea to accept an answer so quickly. This is especially true for wide-open questions such as above. I'd recommend not accepting any answer for at least a month so that it wil get exposed to as many members as possible. – Bill Dubuque Dec 29 '11 at 17:20
• Dubuque, thank you for your tip. – MathOverview Dec 29 '11 at 22:11

If an abstract category-theoretic answer will satisfy you, then you could say something very general like

A symmetry of a morphism $\phi:A\to B$ means a pair $(\alpha,\beta)$ of automorphisms of $A$ and $B$ respectively, such that $\beta \circ \phi = \phi \circ \alpha$.

One easily sees that the symmetries of any $\phi$ constitute a group. If $\phi$ is mono (or epi) then $\beta$ determines $\alpha$ (or vice versa), and the symmetry group is a subgroup of $\operatorname{Aut}(B)$ (or $\operatorname{Aut}(A)$).

As a special case, when $\phi$ is an identity morphism, a symmetry is just an automorphism.

Ordinary geometric symmetries arise in this framework in the category of metric spaces and distance-preserving maps, where $\phi$ is the inclusion map from $A\subseteq \mathbb R^n$ into $\mathbb R^n$.

The question is somewhat ill-posed (maybe because it's not entirely clear the example you gave)... At the best of my knowledge, I would naively define a simmetry as a sentence which can be posed in the form "A is like B, seen backwards".

A little algebraic example (maybe not the best): Any ring $R$ corresponds to another ring $R^{op}$, which is the same set with twisted multiplication: if in $R$ multiplication is $(a,b)\mapsto a\cdot b$, in $R^{op}$ multiplication is $(a,b)\mapsto b\cdot a$.

To give you more subtle examples I would like to know how much background you have in geometry, abstract algebra, etc. just in order not to overwhelm you :)

• I believe that the sample is only summarized. It is only a reference to known facts. No explanation of these results. Maybe if we think about symmetry groups polignos regler centered at the origin becomes clearer. And I agree with you that the commutativity is an example of symmetry. – MathOverview Dec 29 '11 at 15:07
• Sorry but I don't understand much of your comment... – Fosco Dec 29 '11 at 16:24