I just came across a strange property of morphisms that are preserved under pullbacks, and it made me wonder. Consider a model category $\mathcal{M}$.
Because the fibrations are exactly the maps that have the right lifting property (with respect to the acyclic cofibrations), it is easy to see that they are closed by pullbacks. Moreover, monomorphisms are also closed by pullbacks. Similarly, both cofibrations and epimorphisms are closed by pushouts.
However, in my mind, I always thought that cofibrations and monomorphisms would be closed under the same type of diagrams, and dually that fibrations and epimorphisms would be closed under the same type of diagrams. I realize that its silly and it may be dangerous to associate cofibrations to monomorphisms, but this association is more of a "mental trick to remember things" than anything.
I don't know if this is a reasonnable question but here I go :
Is there a more rational explanation about why cofibrations and epi are "together", and similarly for fibrations and mono ?
For example on Set, I think there is a model structure in which cofibrations are epi and fibrations are mono. The factorization of $X \stackrel{f}{\to} Y$ would be through the image $X \stackrel{f}{\twoheadrightarrow} \text{im}(f) \hookrightarrow Y$, and so, this forces mono to be closed under pullbacks and epi under pushouts, since fibrations and cofibrations are.
Is there such a model on any complete and cocomplete category for example ?
Thank you. Bogdan