6
$\begingroup$

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.

$\endgroup$
  • 2
    $\begingroup$ 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... $\endgroup$ – Simon Henry Feb 17 at 15:30
  • $\begingroup$ But maybe full accessible image of accessible categories might a good accessibility condition... $\endgroup$ – Simon Henry Feb 17 at 20:21
  • $\begingroup$ 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 ? $\endgroup$ – Simon Henry Feb 17 at 21:33
  • $\begingroup$ @SimonHenry I'm not sure. Maybe you're right! $\endgroup$ – Tim Campion Feb 18 at 1:33
  • 2
    $\begingroup$ 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. $\endgroup$ – Jiří Rosický Feb 18 at 7:10

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.