equivalences in/of $(\infty,1)$-categories
For $n \in \mathbb{N}$ a morphism $f \colon X \to Y$ in an (infinity,1)-category is an $n$-monomorphism equivalently if
in its n-image factorization
the first morphism is an equivalence in an (infinity,1)-category
it is an (n-2)-truncated morphism;
Similarly a function $f \colon X \to Y$ in homotopy type theory is an $n$-monomorphism if its $n$-image factorization is via an equivalence in homotopy type theory.
The dual concept is that of n-epimorphism.
$0$-monomorphism are precisely the equivalences.
Every morphism is an $\infty$-monomorphism.
1-monomorphisms are often just called monomorphisms in an (∞,1)-category. The 1-monomorphisms into a fixed object are called the subobjects of that object.
A 1-monomorphism between 0-truncated objects is precisely an ordinary monomorphism in the underlying 1-topos.