extension in nLab

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### Context
#### Category theory
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[[!include category theory - contents]]
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> This entry is about two senses of _extension_: extension of *[[morphisms]]*, [[formal dual|dual]] to [[lift]], and extension of [[objects]] ([[algebra extension]]).  In [[foundations]] and [[formal logic]]  there is also *[[extension (semantics)|semantical extension]]* and *[[context extension]]*.


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## Idea

### Extension of morphisms
 {#ExtensionOfMorphisms}

An **extension** 

* of a [[morphism]] $f: A\to Y$ 

* along a morphism $i \colon A\hookrightarrow X$ 

  (often assumed to be a [[monomorphism]]) 

is 

* a morphism $\tilde{f} \colon X\to Y$ 

* such that $\tilde{f}\circ i = f$, 

hence a completion of the [[span]] to a [[commuting diagram]] like this:

\begin{tikzcd}
  A
  \ar[r, "{ f }"]
  \ar[d, "{ i }"]
  &
  Y
  \\
  B
  \ar[
    ur, 
    dashed,
    "{ \widetilde f }"{swap}
  ]
\end{tikzcd}


The [[formal duality|dual]] problem is that of [[lifting]] a morphism $f \colon Y\to B$ through an ([[epimorphism|epi]])morphism $p \colon X\to B$, giving a morphism $\tilde{f}: Y\to X$ such that $f = p\circ\tilde{f}$.



### Extension of an object by another object 

In a [[category]] $C$ with a notion of [[short exact sequence]] (e.g. any [[semiabelian category]], [[Quillen exact category]] etc.) an **extension** of an object $Q$ by an object $K$ is any object $X$ fitting in a [[short exact sequence]] of the form

$$ 
  K 
  \stackrel{i}
  \to
  X 
  \stackrel{p}
  \to 
  Q
  \,.
$$

For further cases, such as _[[group extension]]_, _[[Lie algebra extension]]_, _[[infinitesimal extension]]_ etc., see at _[[algebra extension]]_. 


Classification of extensions in many categories is obtained using a [[forgetful functor]] $C\to D$ to a simpler category $D$, which preserves short exact sequences. For example, if all extensions in $D$ are isomorphic to $K\coprod Q$, then one looks for an additional structure in $C$ needed to equip the coproduct $K \coprod Q$ with a structure of an object in $C$ such that the $i$ and $p$ are morphisms in
$C$ making above a short exact sequence in $C$.

### Extension of an object as an enlargement 

In something like the [[dual]] sense of the above, *extension* is also used to indicate a kind of enlargement of an object in the sense that the object being extended can be viewed as a [[subobject]] of the extension. This is the case for [[field extensions]], $k \hookrightarrow K$.



### Other notions of extension

* [[extension (semantics)]]

* [[context extension]]

* [[absolute extensor]]

* [[extension of distributions]]


## Examples 

### Extension of functions

The [[Tietze extension theorem]] is about *extensions of continuous maps* from a subspace to a normal toplogical space. 

[[!include extension theorems -- table]]

### Exension of differential forms

See *[[extension lemma for differential forms]]*.

### Extension of representations

See at *[[extended representation]]*,

### Group extensions

For example, in the category [[Grp]] of (possibly nonabelian) groups one has a short exact sequence usually denoted $1\to X\to Z\to Y\to 1$ corresponding to a [[group extension]].

## Related concepts

* [[homotopy extension property]]

* [[Kan extension]]

* [[lifts and extensions]]

## References

General discussion with an eye towards [[algebraic topology]] and the [[Tietze extension theorem]]:

* [[Norman Steenrod]], _Cohomology operations, and obstructions to extending continuous functions_, Advances in Mathematics Volume 8, Issue 3, June 1972, Pages 371-416 (<a href="https://doi.org/10.1016/0001-8708(72)90004-7">doi:10.1016/0001-8708(72)90004-7</a>, [pdf](http://www.maths.ed.ac.uk/%7Eaar/papers/steen6.pdf))



[[!redirects extensions]]