Definition:Isomorphism (Category Theory)

Definition
Let $\mathbf C$ be a category, and let $X, Y$ be objects of $\mathbf C$.

A morphism $f: X \to Y$ is an isomorphism if there exists a morphism $g: Y \to X$ such that:


 * $g \circ f = \operatorname{id}_X$
 * $f \circ g = \operatorname{id}_Y$

where $\operatorname{id}_X$ denotes the identity morphism on $X$.

It can be seen that this is equivalent to $g$ being a retraction and section of $f$.

Also known as
Some authors, to avoid tedium, speak simply of an iso.

Furthermore, in place of the consistent phrasing "$f$ is an iso" they will generally prefer the shorter "$f$ is iso".

Linguistic Note
The word isomorphism comes from the Greek morphe (μορφή) meaning form or structure, with the prefix iso- meaning equal.

Thus isomorphism means equal structure.