There are no noncommutative soft maps

TL;DR

This paper characterizes when the induced unital *-homomorphism from a map between compact spaces is projective, showing it only occurs for dendrites with homeomorphism or constant maps, revealing a noncommutative analogue of soft maps.

Contribution

It establishes a precise characterization of projective *-homomorphisms in the noncommutative setting for maps between compact spaces, focusing on dendrites.

Findings

Unital *-homomorphism is projective iff the space is a dendrite and the map is a homeomorphism or constant.

Provides a noncommutative analogue of soft maps in topology.

Clarifies the structure of projective maps in the category of C*-algebras associated with compact spaces.

Abstract

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