Injectivity of Lipschitz operators

Any Lipschitz map $f\colon M \to N$ between metric spaces can be "linearised" in such a way that it becomes a bounded linear operator $\widehat{f}\colon \mathcal F(M) \to \mathcal F(N)$ between the Lipschitz-free spaces over $M$ and $N$. The purpose of this note is to explore the connections between the injectivity of $f$ and the injectivity of $\widehat{f}$. While it is obvious that if $\widehat{f}$ is injective then so is $f$, the converse is less clear. Indeed, we pin down some cases where this implication does not hold but we also prove that, for some classes of metric spaces $M$, any injective Lipschitz map $f\colon M \to N$ (for any $N$) admits an injective linearisation. Along our way, we study how Lipschitz maps carry the support of elements in free spaces and also we provide stronger conditions on $f$ which ensure that $\widehat{f}$ is injective.