The (epi,mono) factorization system in Sets is part of a model structure on Sets whose weak equivalences are the epis, fibrations are monos and cofibrations are everything. This is a model for the homotopy theory of (-1)-truncated sets. One can also define a similar model structure on simplicial sets, where the weak equivalences are the maps which are surjective on pi_0 (i.e, homotopy epis) and whose fibrations are the Kan fibrations whose homotopy fibers are all (-1)-truncated (i.e., homotopy monos). This is a also a model for (-1)-truncated spaces/sets. A similar thing can be done in others contexts as well, for example, you can replace simplicial sets by a model category which presents some $\infty$-topos, replace weak equivalences with homotopy epis and fibrations with fibrations which are homotopy monos. The new model category will present the homotopy theory of (-1)-truncated objects in the original $\infty$-topos. I will go on a limb here and suggest that the (epi,mono) type factorization systems, which appear very typical from a 1-categorical point of view, are, from a model categorical point of view, essentially the particular case of (well-behaved) (-1)-truncated homotopy theories.

The (epi,mono) factorization system in Sets is part of a model structure on Sets whose weak equivalences are the epis, fibrations are monos and cofibrations are everything. This is a model for the homotopy theory of (-1)-truncated sets. One can also define a similar model structure on simplicial sets, where the weak equivalences are the maps which are surjective on pi_0 (i.e, homotopy epis) and whose fibrations are the Kan fibrations whose homotopy fibers are all (-1)-truncated (i.e., homotopy monos). This is a also a model for (-1)-truncated spaces/sets. A similar thing can be done in others contexts as well, for example, if you can replace simplicial sets by a model category of simplicial presheaves which presents a certain $\infty$-topos, then you can often left Bousfield localize $M$ such that the new weak equivalences are homotopy epis and the fibrations (at least between old fibrant objects) are the old fibrations which are also homotopy monos. The new model category will present the homotopy theory of (-1)-truncated objects in the original $\infty$-topos. I will go on a limb here and suggest that the (epi,mono) type factorization systems, which appear very typical from a 1-categorical point of view, are, from a model categorical point of view, essentially the particular case of (certain kinds of) (-1)-truncated homotopy theories.