# Definition:Isomorphism (Category Theory)

*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 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**.

## Also see

## Sources

- 2010: Steve Awodey:
*Category Theory*(2nd ed.) ... (previous) ... (next): $\S 1.5$: Definition $1.3$