A recent question [Why do we need model categories?](https://mathoverflow.net/q/287091/41291) reminded me of this long-standing confusion of mine -- I mentioned it in an answer there, and then decided to ask a separate question about it. I even dare not to use the [soft-question](/questions/tagged/soft-question "show questions tagged 'soft-question'") tag.


What confuses me is this. There is an obvious resemblance between model categories and factorization systems. Yet, one thing goes totally wrong: in factorization systems, "left halves" tend to be "like epimorphisms" and "right halves" - "like monomorphisms". At the same time, in model categories cofibrations have distinct flavor of monos to them, and fibrations - of epis.


You see, I cannot even formulate this rigorously, yet I hope you agree that what I now described contains undeniable truth.


So what is this truth? Does this phenomenon have any explanation? Living with this puzzle for many years, the only consideration that I've been able to come up with is this: take one step from sets to categories. In sets, monos are just inclusions. In categories, the "correct" notion of mono starts to involve some amount of "epiness", since "good" monos in categories are **full** and faithful functors, and fullness involves some surjectivity condition. I don't even know whether there is some sort of such "first-step-consideration" from the opposite end, relating "betterness" of epis with some sort of additional requirements which have to do with monomorphy.


I repeat - this is a strange question: although it is full of most vague handwaving, I hope you agree that it touches on something very rigorous, which I just fail to capture.



The cofibration/fibration $\leftrightarrow$ epi/mono confusion
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.


