When left inverse of a function is injective

Question

Consider function $f^{-1}$ which is a left inverse of another function $f$. I require that $f^{-1}$ must be injective. What does it tell me about $f$? In other words, can I put some constraints on $f$ (for example also being injective) which would guarantee that $f^{-1}$ exists and is injective?

Answer 1

Consider functions $g:Y \leftarrow X$ and $f:X \leftarrow Y$.

Then from $g \circ f = \mathrm{id}_Y$, we can deduce that $g$ is surjective and $f$ is injective. The existence of such a $g$ (called a retraction of $f$, or simply a left-inverse) is equivalent to the condition "$Y$ is non-empty and $f$ is injective." To answer your question: the existence of an injective retraction is equivalent to the condition "$f$ is bijective."