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One of the axioms for a model structure on a category $C$ is that any morphism $f$ of $C$ can be factored as $f = pj$, where $p$ is a fibration, $j$ is a cofibration, and we can choose either one to be a weak equivalence.

In general I have the impression that in some sense cofibrations are analogous to injections and fibrations are analogous to surjections. Some of the nice properties coincide, also the notation for the arrows makes you think that, and sometimes when we construct a model structure we impose some condition on cofibration like "being injective on objects": think the model structure for the categories of simplicial sets or small categories.

However, when it comes to sets, the factorization into an injection followed by a surjection is quite unnatural. On the other hand, the factorization into a surjection followed by the injection is just factoring through the image, which is extremely natural. So how come did we define the factorization axiom this way and not the other way around?

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    $\begingroup$ Because that's how fibrations and cofibrations work in homotopy theory? $\endgroup$ Apr 18 '18 at 6:07
  • $\begingroup$ Actually, factoring a map $f\colon A\to B$ into the inclusion of its graph $(id,f)\colon A \to A\times B$ and the projection $p_B\colon A\times B \to B$ looks quite natural. $\endgroup$ Oct 2 '18 at 14:27
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I don't think it's very accurate to think of fibrations as "surjections" and cofibrations as "injections". For one thing, in the familiar topological spaces/simplicial set model structures, fibrations don't have to be surjective! For instance, the unique map from the empty space to any space is a fibration.

A better analogy is that fibrations are maps that are like "fiber bundles" or otherwise "nice locally on the codomain" and have nice lifting properties. Cofibrations, on the other hand, are like injections obtained by "adjoining nice pieces" to the domain. (For instance, Serre cofibrations are maps $X\to Y$ obtained by adjoining cells, and retracts of such maps.)

In particular, if you start with an arbitrary map $f:X\to Y$, it's not reasonable to expect to factor it through a fibration $X\to Z$ or a cofibration $Z\to Y$. This would require $X$ to be a nice "fiber bundle-like" space over a base $Z$, but if $X$ is some arbitrary object this may not be possible in any nontrivial way at all. Similarly, a cofibration $Z\to Y$ would require $Y$ to be "built nicely" from $Z$, but if $Y$ itself is not nice at all, it may not be possible to build it nicely from another space.

On the other hand, getting a cofibration $X\to Z$ and a fibration $Z\to Y$ is a much more realistic goal. You can get cofibrations coming out of any object by just adding nice pieces to it. And you can get a fibration over any object by taking natural fiber bundle-like constructions. (Or, more relevantly to how one actually proves these factorization properties, by a "small object argument" that exhaustively adds pieces to $Z$ until the required lifting property holds.)

Another way to think about it is that these factorizations are like souped-up versions of something very classical: given a map $f:X\to Y$ of topological spaces, you can factor it through the mapping cylinder. The mapping cylinder is homotopy equivalent to $Y$, but $X$ injects nicely into it, and so by replacing $Y$ with the mapping cylinder you can assume (up to homotopy equivalence) that the map $f$ is just a nice injection. The factorization of $f$ as a cofibration followed by an acyclic fibration is the same idea, but even nicer: you turn $f$ into a nice inclusion (cofibration) by replacing $Y$ with a weak equivalent object, and your weak equivalence is also a fibration! (Dually, the other factorization is similar to the factorization of a map of spaces through the mapping path space.)

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