# Characterizing the left / right classes of (weak) factorization systems in locally presentable categories

Let $$\mathcal M \subseteq Mor(\mathcal C)$$ be a class of morphisms in a locally presentable category.

1. It's well-known that $$\mathcal M$$ is the left half of an accessible orthogonal factorization system iff $$\mathcal M$$ is accessible (as a full subcategory of the morphism category $$\mathcal C^{[1]}$$), and is closed under colimits in $$\mathcal C^{[1]}$$, cobase-change, composition, and isomorphisms.

2. Analogously (though perhaps this is less well-known), $$\mathcal M$$ is the right class of an accessible orthogonal factorization system iff it is accessible and accessibly-embedded in $$\mathcal C^{[1]}$$, and closed under limits in $$\mathcal C^{[1]}$$, base-change, composition, and isomorphisms. The proof of course is not dual -- one observes that under these conditions, $$\mathcal M$$ is accessibly-reflective in $$\mathcal C^{[1]}$$, shows that one leg of each unit map must be an isomorphism, so that the reflector provides factorizations, and then verifies a few things.

In the case of weak factorization systems (wfs), the situation can't be quite so simple. For one thing, not all accessible wfs on a locally presentable category are cofibrantly-generated, so any "small generation" argument is going to be more delicate.

1. More to the point, the left class of a wfs can be accessible without the wfs being accessible, and conversely (at least under anti-large-cardinal hypotheses) the left class of an accessible wfs need not be accessible. So even though one might guess that closure under coproducts, cobase-change, isomorphisms, composition, transfinite composition, and retracts should nearly characterize left classes of accessible wfs on locally presentable categories, it's not clear what kind of "accessibility" hypothesis to add to get a characterization.

2. Nevertheless, there might be more hope for characterizing the right classes of wfs on locally presentable categories. In particular, the following guess seems reasonable:

Question: Let $$\mathcal M \subseteq Mor(\mathcal C)$$ be a class of morphisms in a locally presentable category. Suppose that $$\mathcal M$$ is accessible and accessibly embedded in $$\mathcal C^{[1]}$$, and closed under products, base change, isomorphisms, composition, co-transfinite composition, and retracts. Does it follow that $$\mathcal M$$ is the right class of an accessible weak factorization system on $$\mathcal C$$? If not, is there a characterization along similar lines?

Presumably the proof of any characterization will proceed by using some form of Garner's small object argument, but beyond that it's unclear to me.

Actually, there's a stronger condition than closure under dual transfinite composition which might be needed: say that a morphism in $$\mathcal M \subseteq \mathcal C^{[1]}$$ (corresponding to a commutative square in $$\mathcal C$$ with two opposite legs lying in $$\mathcal M$$) is $$\mathcal M$$-cartesian if the comparison map into the pullback lies in $$\mathcal M$$. If $$\mathcal M$$ is the right class of a wfs, I believe it's the case that the wide subcategory of $$\mathcal M$$ whose morphisms are the $$\mathcal M$$-cartesian squares is closed in $$\mathcal C^{[1]}$$ under cofiltered limits. We might have to include this condition in our putative characterization.

• I'm very interested by this question. I might be something though: is it clear that right class of accessible weak factorization are accessible and accessibly embedded ? They are full images of accessible functors, but as far as I can tell this is their only accessibility property that I know of... – Simon Henry Feb 17 at 15:30
• But maybe full accessible image of accessible categories might a good accessibility condition... – Simon Henry Feb 17 at 20:21
• Some care is needed, but I feel like full accessible image might really be an interesting condition for left classes. If you have a left class that is a full image of a $\lambda$-accessible functor $F: V \rightarrow C^\rightarrow$ it sounds tempting to consider the algebraic weak factorization system generated by $V_\lambda$... I would have to think more about it, but it sounds like a promising strategy... Unless you already know this does not work ? – Simon Henry Feb 17 at 21:33
• @SimonHenry I'm not sure. Maybe you're right! – Tim Campion Feb 18 at 1:33
• Both the left and the right class of an accessible wfs are full images of an accessible functor. Assuming the existence of a proper class of almost strongly compact cardinals, they are preaccessible. Consider a wfs $(Emb,Top)$ in $\bf{Pos}$. Here $Emb$ is accessible and, if this wfs is accessible, complete lattices (= $Emb$-injectives) are a full image of an accessible functor. Assuming the existence of a proper class of almost strongly compact cardinals, complete lattice are accessibly embedded in posets, which is not true. Thus, under this assumption, $(Emb,Pos)$ is not accessible. – Jiří Rosický Feb 18 at 7:10