What is the example of non canonical isom?

Question

In mathematics, isomorphism $f$ between two objects are sometimes called canonical when $f$ is unique as a map, in other words, everyone choose the same map, no different map gives the isomorphism.

Could you give me an example of non canonical map, and explain why that is not canonical ?

Thank you in advance.

Answer 1

$\DeclareMathOperator{\GL}{GL}$I am not so sure about your definition of canonical. As I understand it, it is an isomorphism that does not depend on a particular choice.

The standard example is given by a finite dimensional vector space $V$ and its dual $V^{\star}$. An isomorphism between the two depends on the choice of a base on $V$. (Addendum. The point should be that an element $0 \ne f \in V^{\star}$ is determined by its kernel $K$, which is a hyperplane of $V$, and then a vector $v \notin K$ such that $f(v) = 1$. But even when $K \ne \{ 0 \}$ is given, there are many such vectors, as $f(v) = 1$ implies $f(v + k) = 1$ for $k \in K$, so there is no way to associate in a base-independent way an element of $V^{\star}$ to a vector of $V$.)

The situation is different with $V$ and its bidual $V^{\star\star}$, as the map $$ v \mapsto (f \mapsto f(v)) $$ is an isomorphism that does not depend on any particular choice. But there are of course many other isomorphisms, just compose this one with an element of $\GL(V)$.

Answer 2

I think that the words are used slightly differently in different areas. @Andreas Caranti has given you the standard example of how the words are used in linear algebra, essentially to mean "defined without reference to any particular basis".

In group theory I think the words are used in a weaker sense.

As you have tagged this "groups" here is a non-canonical isomorphism between $A_4/V$ and $A_3$. (Here $A_3$, $A_4$ are the alternating groups of orders $3$ and $12$ and $V$ is the normal subgroups of order $4$ in $A_4$).

Define $f:A_3\to A_4/V$ by $f(e)=V$, $f((123))=(132)V$, $f((132))=(123)V$. You can check that this is an isomorphism. However the canonical isomorphism - the one "we would all choose" - does not swap the $3$-cycles in this way.