## Do you really mean S^1?

This is a follow up post to my previous post on $\mathbb{R}^n$. Mathematicians will often write $S^1$ without being clear of the context and structure associated with it.

To a Euclidean geometer $S^1$ means a circle – a maximal set of points equidistant from a given point. All circles are equivalent in the sense that they can be made equal by a translation and a scaling about the centre.

In a two dimensional inner product space $S^1$ typically means the set of all points with norm 1. A circle is more generally the set of all points with norm r for some real number r and is related to $S^1$ by a scaling transformation.

To a group theorist $S^1$ would mean the one dimensional orthogonal group – the group of all transformations in the plane.

To a complex analyst $S^1$ could mean either the set of points in $\mathbb{C}$ with length 1 or it could be the group of linear transformations associated with multiplication by elements of this set. (These are quite different – in the set 1 has no special meaning but in the group it corresponds to the identity).

To a topologist $S^1$ means anything homeomorphic to a Euclidean circle – so includes ellipses, polygons, simple closed curves,…

To a differential geometer $S^1$ means anything diffeomorphic to a Euclidean circle, which doesn’t include “most” things a topologist means.

A set theorist wouldn’t call it $S^1$, but to her it would be any set with the same cardinality as the real numbers.

This is a major theme of category theory – it’s not only the objects that matter but also the maps that preserve them  – whether it be affine transformations, orthogonal transformations, group homomorphisms, group homomorphisms, homeomorphisms, diffeomorphisms or bijections. So if you must write $S^1$ to represent a structure at least make sure it is clear which category you are working in – i.e. which maps preserve the structure.