There are no noncommutative soft maps

Abstract

It is shown that for a map $f:X\to Y$ of compact spaces the unital $\ast$-homomorphism $C(f):C(Y)\to C(X)$ is projective in the category $\mathrm{Mor}({\mathcal{C}}^1)$ precisely when $X$ is a dendrite and $f$ is either homeomorphism or a constant.