Isomorphism

A pair of mathematical structures are isomorphic to each other if they are "essentially the same", even if they aren't necessarily equal.

An isomorphism is a between isomorphic structures which translates one to the other in a way that preserves all the relevant structure. An important property of an isomorphism is that it can be 'undone' by its isomorphism.

An isomorphism from an object to itself is called an . They can be thought of as symmetries: different ways in which an object can be mapped onto itself without changing it.

Equality and Identity

The simplest isomorphism is equality: if two things are equal then they are actually the same thing (and so not actually two things at all). Anything is obviously indistinguishable from itself under whatever measure you might use (it has any property in common with itself) and so regardless of the theory or language, anything is isomorphic to itself. This is represented by the (iso)morphism.

Isomorphisms in Category Theory

In , an isomorphism is a morphism which has a two-sided . That is to say, $f:A\to B$ is an isomorphism if there is a morphism $g:B\to A$ where $f$ and $g$ cancel each other out.

Formally, this means that both composites $fg$ and $gf$ are equal to identity morphisms (morphisms which 'do nothing' or declare an object equal to itself). That is, $gf={id}_A$ and $fg={id}_B$.