An asymptotic homotopy lifting property

A $C^{\ast }$-algebra $A$ is said to have the homotopy lifting property if for all $C^{\ast }$-algebras $B$ and $E$, for every surjective ${}^{\ast }$-homomorphism $\pi \text{\hspace{0.17em}⁣}:E\to B$ and for every ${}^{\ast }$-homomorphism $\phi \text{\hspace{0.17em}⁣}:A\to E$, any path of ${}^{\ast }$-homomorphisms $A\to B$ starting at $\pi \phi$ lifts to a path of ${}^{\ast }$-homomorphisms $A\to E$ starting at $\phi$. Blackadar has shown that this property holds for all semiprojective $C^{\ast }$-algebras. We show that a version of the homotopy lifting property for asymptotic morphisms holds for separable $C^{\ast }$-algebras that are sequential inductive limits of semiprojective $C^{\ast }$-algebras. It also holds for any separable $C^{\ast }$-algebra if the quotient map $\pi$ satisfies an approximate decomposition property in the spirit of (but weaker than) the notion of quasidiagonality for extensions.