Examples of non-invariant yet "useful" properties of mathematical objects

Question

I am trying to find out whether there are mathematically important or useful properties (of some object(s)) that are nevertheless not invariant under some usual choice of isomorphism?

Are there any such "natural" examples of objects having some property (expressed in some language) which are non-invariant?

What I mean by "natural" is roughly that I want a non ad-hoc example, i.e. something actually used/known by some mathematician(s) that they find important for some purpose or other.

The background to my questioning is that I would like to probe whether some of the claims made by structuralist philosophers of mathematics, specifically that "structural" properties are the only mathematically relevant properties, hold any water. By structural properties I mean properties invariant under isomorphisms in some category (in whatever relevant sense).

So far, I have not been able to find a convincing counter-example (one that isn't ad-hoc) to the above structuralist claim, hence my question.

Answer 1

In setting up the categories of Banach spaces (or Hilbert spaces or inner product spaces or normed spaces), we choose to take continuous linear mappings as the morphisms rather than isometric linear mappings. The metric structure is not invariant under this choice for the morphisms, but the resulting categories are far more interesting and useful.

Answer 2

An elliptic curve is said to be in Weierstrass normal form if it is defined by an equation of the form $$ y^2 = x^3 + ax + b $$ for constants $a,b$. This property is not preserved under automorphisms of elliptic curves: in fact every elliptic curve over a field of characteristic not 2 or 3 is isomorphic to an elliptic curve in Weierstrass form. Being in Weierstrass normal form is therefore not an intrinsic property of the elliptic curve itself, but of the presentation of the elliptic curve. The reason that Weierstrass normal forms are useful, despite not being preserved under isomorphism, is that it simplifies some calculations.

Something similar happens for matrices over $\mathbb C$. Every square matrix over $\mathbb C$ is similar to a square matrix in Jordan normal form. Clearly, the property of `being in Jordan normal form' is not preserved under similarities, which is the natural notion of isomorpism for square matrices in most situations. Again, the reason that we are studying the Jordan normal form (and similar decompositions) anyway, is that it helps when doing explicit calculations.