The category of fibrations in a combinatorial model category is accessible, accessibly embedded in the arrow category. How about the cofibrations?


More generally, let $C$ be a locally presentable category and let $(L,R)$ be a weak factorization system on $C$. If $(L,R)$ is cofibrantly-generated (i.e. there is a set $I \subseteq L$ such that $R$ consists precisely of those morphisms with the right lifting property with respect to $I$), then $R$, considered as a full subcategory of $C^\to$, is accessible and accessibly embedded.


**Question 1:** Suppose that $(L,R)$ is cofibrantly-generated. Is $L$ accessible and accessibly embedded (as a full subcategory of $C^\to$)?


**Question 2:** Conversely, if $L$ is accessible and accessibly embedded, then is $(L,R)$ cofibrantly-generated?


**Question 3:** Similar to the above two, but use the notion of "small-generated" coming from Garner's small object argument (where $I$ can be a category rather than a set).


The proof that $R$ is accessible and accessibly embedded is not completely straightforward: it relies on the fact that the small object argument provides a functorial factorization system which preserves $\lambda$-filtered colimits and $\lambda$-presentable objects for some $\lambda$ to exhibit every $R$-morphism as a retract of a colimit of $\lambda$-presentable $R$-morphisms and to see that fibrations are closed under $\lambda$-filtered colimits.


The fact that $L$ is closed under transfinite composition sounds tantalizingly close to saying that it is closed under filtered colimits, but I'm not sure the latter is actually true.


**Motivation:**
If the answer to both questions is *yes*, then it becomes very easy to prove Jeff Smith's theorem since an intersection of accessible, accessibly-embedded, replete subcategories is accessible and accessibly-embedded.



Are cofibrations accessible?
A cofibrantly generated $(L,R)$ does not need to have $L$ accessible, see Example 3.5
in my paper "On combinatorial model categories."
Also, $L$ accessible does not imply that $(L,R)$ is cofibrantly generated, even accessible.
Take regular monos in Boolean algebras. This $L$ is accessible but $(L,R)$ cannot be accessible because
regular injectives are complete Boolean algebras which are not accessible.


