lifts and extensions



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.


Extension problem

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

Retraction problem

Let i:AXi:A\to X be a morphism. Find a retraction of ii, that is a morphism r:XAr:X\to A such that ri=id Ar\circ i = id_A.

The retraction problem is a special case of the extension problem for Y=AY=A and f=id Af=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 AXY\coprod_A X exists (for ii, ff as above) then the extensions f˜\tilde{f} of ff along ii are in 1–1 correspondence with the retractions of i *(f):YY AXi_*(f) : Y\to Y\coprod_A X.

Lifting problem

Given morphisms p:EBp:E\to B and g:ZBg:Z\to B, find a lifting of gg to EE, i.e. a morphism g˜:ZE\tilde{g}:Z\to E such that pg˜=gp\circ\tilde{g}=g.

Section problem

For any p:EBp:E\to B find a section s:BEs: B\to E, i.e. a morphism ss such that ps=id Bp\circ s = id_B.

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

Proposition (Reducing a lifting to a section)

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

category: disambiguation