nLab lifts and extensions

Idea

Given any morphism in some category a basic question is to ask for lifts/extensions along it (which are dual notions), and in particular for retracts/sections.

We survey how these concepts relate to each other. See the respective entries for more details and pointers.

Definitions

Extension problem

Given morphisms $i:A\to X$, $f:A\to Y$ find an extension of $f$ to $X$, i.e. a morphism $\tilde{f}:X\to Y$ such that $\tilde{f}\circ i=f$. Notice that if $i:A\hookrightarrow X$ is a subobject, then $\tilde{f}\circ i$ is the restriction $\tilde{f}{|_A}$, and the condition is $\tilde{f}{|_A} = f$.

Retraction problem

Let $i:A\to X$ be a morphism. Find a retraction of $i$, that is a morphism $r:X\to A$ such that $r\circ i = id_A$.

The retraction problem is a special case of the extension problem for $Y=A$ and $f=id_A$. Conversely, the general extension problem may (in Top and many other categories) be reduced to a retraction problem:

Proposition (Reducing an extension to a retraction)

If the pushout $Y\coprod_A X$ exists (for $i$, $f$ as above) then the extensions $\tilde{f}$ of $f$ along $i$ are in 1–1 correspondence with the retractions of $i_*(f) : Y\to Y\coprod_A X$.

Lifting problem

Given morphisms $p:E\to B$ and $g:Z\to B$, find a lifting of $g$ to $E$, i.e. a morphism $\tilde{g}:Z\to E$ such that $p\circ\tilde{g}=g$.

Section problem

For any $p:E\to B$ find a section $s: B\to E$, i.e. a morphism $s$ such that $p\circ s = id_B$.

The section problem is a special case of a lifting problem where $g = id_B : B\to B$. Then the lifting is the section: $\tilde{g} = s$. A converse is true in the sense

Proposition (Reducing a lifting to a section)

If the pullback $Z\times_B E$ exists then the general liftings for of $G$ along $p$ as above are in a bijection with the section of $g^*(p)=Z\times_B p : Z\times_B E\to Z$.

Related concepts

• lift/extension

• retract/section

category: disambiguation