A space $Y$ is called an *absolute extensor for normal spaces* (also sometimes *solid*) if, for any normal space $X$, closed subset $A$ of $X$, and map $f:A\to Y$, there exists a map $f′:X\to Y$ such that $f′|A=f$, i.e.
$A\to X$ has the left lifting property with respect to the map $Y\to pt$
from $Y$ to a singleton $pt$
$$A\to B \perp Y\to pt$$

~~What is the analogous notion for a ~~ The analogous notion for maps instead of spaces is called being *map* $g:Y_1\to Y_2$ instead of a space $Y$?*soft with respect to a pair of spaces $(A,B)$* where $A$ is a closed subset of a normal space $B$. (E. V. Shchepin, Soft maps of manifolds, Uspekhi Mat. Nauk, 1984, Volume 39, Issue 5(239), 209–224; \S2, Def.).
Are there any references ? Obviously a necessary condition is being a Serre acyclic fibration.

In more details:

Scchepin [ibid] calls a map $g:Y_1\to Y_2$

soft with respect to a pair of spaces $X,A$where $A\subset X$ iff the inclusion map $i:A\to X$ has the left lifting property with respect to the map $Y\to pt$ from $Y$ to a singleton $pt$ $$A\xrightarrow{i} B \perp Y_1\xrightarrow{g} Y_2$$

What is known about this notion ?

Is a Serre fibration of sufficiently nice spaces (say, a cellular map of finite CW complexes) necessarily soft for any pair $X,A$ where $A$ is a closed subset of hereditary perfectly normal space $X$ ?

A map $p:|Y|\to Y$ from the geometric realisation of a finite simplicial complex to the simplicial complex viewed as a finite topological space, does indeed have this property, by the lifting theorem for finite spaces (Thm. 3.5, Eric Wofsey), see details of the statement in this question.

soft mapby Eugene Schepin , e.g. see E. V. Shchepin, Soft maps of manifolds, Uspekhi Mat. Nauk, 1984, Volume 39, Issue 5(239), 209–224, or A. Chigogidze, Inverse Spectra, North Holland, Amsterdam, 1996, and reference therein. I haven't yet understood whether they are related to this particular question. $\endgroup$finitecomplexes, then something should be true. How far you get with this will probably depend on your knowledge of selection theory. $\endgroup$4more comments