structures in a cohesive (∞,1)-topos
The -connected/-truncated factorization system is an orthogonal factorization system in an (∞,1)-category, specifically in an (∞,1)-topos, that generalizes the relative Postnikov systems of ∞Grpd: it factors any morphism through its (n+2)-image by an (n+2)-epimorphism followed by an (n+2)-monomorphism.
As ranges through these factorization systems form an ∞-ary factorization system.
Let be an (∞,1)-topos. For all the class of n-truncated morphisms in forms the right class in a orthogonal factorization system in an (∞,1)-category. The left class is that of n-connected morphisms in .
For this says that effective epimorphisms in an (∞,1)-category have the left lifting property against monomorphisms in an (∞,1)-category. Therefore one may say that the effective epimorphisms in an -topos are the strong epimorphisms.
This appears as (Lurie, prop. 188.8.131.52(6)).
Moreover, every morphism is (-2)-connected.
Therefore for the -connected/-truncated factorization system says (only) that equivalences have inverses, unique up to coherent homotopy.
is full and faithful precisely if it is an injection;
has non-empty fibers precisely if it is an epimorphism.
Therefore on homotopy 1-types the 0-connected/0-truncated factorization system is the (eso+full, faithful) factorization system.
The general abstract statement is in
A model category-theoretic discussion is in section 8 of