equivalences in/of $(\infty,1)$-categories
One analog in (∞,1)-category theory of epimorphism in category theory. Beware that there are other variants such as effective epimorphism in an (infinity,1)-category and generally the concept of n-epimorphism.
For $C$ an (∞,1)-category, a morphism $f : X \to Y$ in $C$ is an epimorphism if for all $A \in C$ the induced morphism
is a monomorphism in an (∞,1)-category in ∞Grpd.
A morphism $A\to B$ of E-infinity rings is an epimorphism iff $B$ is smashing over $A$, i.e., if $B\wedge_A B\approx B$.
A morphis, $X\to Y$ between connected spaces is an epimorphism iff $Y$ is formed via a Quillen-plus construction from a perfect normal subgroup of the fundamental group $\pi_1 X$.