# Definition:Isomorphism (Category Theory)

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This page is about Isomorphism in the context of Category Theory. For other uses, see Isomorphism.

## 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 and only if there exists a morphism $g: Y \to X$ such that:

$g \circ f = I_X$
$f \circ g = I_Y$

where $I_X$ denotes the identity morphism on $X$.

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

### Inverse Morphism

A morphism $g: Y \to X$ is said to be an inverse (morphism) for $f$ if and only if:

$g \circ f = I_X$
$f \circ g = I_Y$

where $I_X$ denotes the identity morphism on $X$.

## 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 derives from the Greek morphe (μορφή) meaning form or structure, with the prefix iso- meaning equal.

Thus isomorphism means equal structure.