# Small object argument for multiple factorization systems

Is there something similar to the small object argument, but related to a chain of factorization systems on a category $\cal C$?

It is easy to see that one can give a chain of "generating morphisms" $J_1\subseteq \cdots \subseteq J_n$ and obtain a reversed chain passing to the orthogonals $$J_1^\perp\supseteq \cdots \supseteq J_n^\perp$$ If each $J_\alpha$ is a set of morphisms with small co/domains this, using the SOA one time for each $J_\alpha$, entails that there is a chain of factorization systems $$\big({}^\perp(J_n^\perp),J_n^\perp \big) \le \cdots \le \big({}^\perp(J_1^\perp),J_1^\perp \big)$$

Series of Bousfield localizations of a given model category are a natural example of this. Is there some reference where this thing is defined from the purely categorical POV (and where, for example, the definition is stated for a transfinite chain $\alpha\to FS(\cal C)$)?

• What is the "small object argument"? What is the problem that you are trying to solve? – Paul Taylor Apr 24 '16 at 8:29
• The small object argument is (roughly) the following statement; let $J\subseteq \hom(\cal C)$ be a set of morphisms, if the domains (or codomains, I never remember) of arrows in $J$ are small (i.e. $\hom(X,-)$ commutes with filtered colimits for each $f\colon X\to Y$ in $J$) then $({}^\perp(J^\perp), J^\perp)$ is a factorization system on $\cal C$. – Fosco Apr 24 '16 at 8:55

I still can't see what the question is. You seem to be asking when a prefactorisation system (an orthogonal pair $(E,M)$ in which both sides are closed in the Galois connection) actually admits factorisation. You need some finite limits or colimits for this, not just filtered ones. Example 5.7.10 in my book (Practical Foundations of Mathematics, CUP 1999) shows a poset with six points that is not a lattice together with a prefactorisation system that is not a factorisation system. It is followed by a version of the special adjoint functor theorem that shows how to get a factorisation system.
• Thanks, I'll have a look; I'm referring to this construction, and in particular to thm. 1 and corollary therein. Smallness assumptions on the domains of maps in $J$ ensure that the prefactorization system is indeed a factorization. The question is: I expect the same construction to be studied in the case of multiple (even transfinite) FS on categories. Has this been done before, in the naive way I described above? – Fosco Apr 24 '16 at 14:14