Terminology: Continuous Surjection/Quotient with Continuous Right-Inverse A continuous map with a continuous left-inverse is called an embedding.  However, what is the correct dual notion?  On one side (if we go from the universal property perspective) then we get the quotient map concept.
However, what is the name of the "dual" object in the sense: a continuous surjection with continuous right-inverse.  Are there any known nice properties of these objects which are analogous to embeddings?
 A: Let's clarify things. Assume $f:X\to Y$. Then if $g:Y\to X$ is such that $f\circ g=id_Y$ then we say $g$ is right inverse of $f$. If $g\circ f=id_X$ then we say $g$ is left inverse of $f$.
With that you are talking about (category theoretic) sections and retractions. A left inverse is retraction, a right inverse is section. On the other hand a map having a left inverse is a section and a map having a right inverse is a retraction.
In topology there are very important special cases of retractions and special cases of sections.
Also note that typically we define an embedding as a homeomorphism onto image. Meaning $f:X\to Y$ is an embedding if the (co)restriction $f:X\to f(X)$ is a homeomorphism. Such map does not have to admit neither left nor right inverse. Your definition is too restrictive. A counterexample is the famous $S^n\to D^{n+1}$, $x\mapsto x$ embedding where $S^n$ is the $n$-dimensional sphere and $D^{n+1}$ is the $n+1$-dimensional disk (as a consequence of the Brouwer fixed point theorem).
