[[Category theory MOC]]
# Extension

Given some $A$ in some [[category]] of algebraic objects, an **extension** $\hat{A}$ is an [[epimorphism]]
$$
\begin{align*}
p: \hat{A} \twoheadrightarrow A
\end{align*}
$$
where typically a [[Kernels and cokernels|kernel]] exists giving the [[short exact sequence]]
$$
\begin{align*}
0 \to K \hookrightarrow \hat{A} \stackrel{p}{\twoheadrightarrow} A \to 0
\end{align*}
$$
and we say $\hat{A}$ is an extension of $A$ by $K$.
In case $p$ is [[Split epimorphism|split epic]], we have a **split extension**.

## Particular examples

- [[Group extension]]
- [[Lie algebra extension]]

## See also

- [[Ring extension]], a different use of the word.

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