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
Best Answer
There is a common explanation. As you already mentioned, in order to show that pullbacks preserve fibrations one does not use all the axioms of the model category structure but only the lifting properties. Given two maps $f\colon A\to B$ and $g\colon X\to Y$, write $f \mathrel{\Box} g$ if every square $$ \matrix{ A & \longrightarrow & X \cr {\scriptstyle f} \big\downarrow {\ } & & {\ } \big\downarrow {\scriptstyle g} \cr B & \longrightarrow & Y } $$ has a (not necessarily unique) diagonal $d\colon B\to X$ making the two resulting triangles commutative.
Now let ${\cal H}$ be a class of maps in a category ${\cal K}$ and define $$ {\cal H}^\Box = \{ g \in {\cal K} \mid \forall h\in {\cal H}: h \mathrel{\Box} g \} $$ and $$ {}^\Box{\cal H} = \{ f \in {\cal K} \mid \forall h\in {\cal H}: f \mathrel{\Box} h \} $$
Then we have that ${\cal H}^\Box$ is always stable under pullbacks and dually ${}^\Box{\cal H}$ is always stable under pushouts.
In a model category with cofibrations ${\cal C}$, fibrations ${\cal F}$ and weak equivalences ${\cal W}$ one has ${\cal F} = ({\cal C}\cap {\cal W})^\Box$.
For the case of monomorphisms, suppose that the underlying category ${\cal K}$ has binary copowers and consider the class ${\cal H} = \{ (id_K|id_K)\colon K+K \to K \mid K \in {\cal K} \}$ of all codiagonal maps. Then ${\cal H}^\Box$ is exactly the class of all monomorphisms of ${\cal K}$
Of course, since the usual proof that pullbacks preserve monomorphisms does not need any assumption about the underlying category, this is a bit overkill.