I read in [nLab](https://ncatlab.org/nlab/show/algebraic+model+category) : *Every [cofibrantly generated model category](https://ncatlab.org/nlab/show/cofibrantly+generated+model+category) structure can be lifted to that of an [algebraic model category](https://ncatlab.org/nlab/show/algebraic+model+category). It is not clear whether or not this is true for any [accessible model category](https://ncatlab.org/nlab/show/accessible+model+category)*. Is there any progress since the most recent edit of this page (January 17, 2019) ?



Does any accessible model category come from an algebraic model category?
For reference, here is a more detailed version of Riehl's argument.


**Definition 1** (Garner, [*Understanding the small object argument*](https://link.springer.com/article/10.1007/s10485-008-9137-4), Proposition 3.8) Let $J$ be a category and $D \colon J \to \mathcal{M}^\to$ a functor. Let $f \colon X \to Y$ be a map in $\mathcal{M}$. A *lifting structure* of $f$ against $D$ consists of a choice of diagonal lift for each object $j$ of $J$ and lifting problem of $D(j)$ against $f$, which is compatible with the morphisms in $J$.


**Theorem 2** (Garner, [*Understanding the small object argument*](https://link.springer.com/article/10.1007/s10485-008-9137-4), Theorem 4.4) Suppose $\mathcal{M}$ is locally presentable. Let $J$ be a small category and $D \colon J \to \mathcal{M}$ a functor. Then there is an algebraic weak factorisation system $(L, R)$ on $\mathcal{M}$ such that $R$-algebra structures on a map $f$ correspond precisely to lifting structures of $f$ against $D$. We say that $(L, R)$ is *algebraically cofibrantly generated by $D$*.


**Lemma 3** Let $D \colon I \to \mathcal{M}$ and $E \colon J \to \mathcal{M}$. Then lifting structures of $f$ against the coproduct $D + E \colon I + J \to \mathcal{M}^\to$ correspond precisely to pairs of lifting structures against $D$ and against $E$.


**Proof** It is obvious how to define the lifting structure against $D + E$, so we just have to check that it satisfies naturality for morphisms in $I + J$. However, the coproduct in categories $I + J$ is disjoint, in the sense that there are no maps $\sigma \colon i \to j$ where $i$ lies in the $I$ component and $j$ in the $J$ component, so naturality does follow from naturality for the lifting structures against $D$ and against $E$.


**Theorem 4** (Rosický, [*Accessible model categories*](https://link.springer.com/article/10.1007/s10485-015-9419-6), Theorem 4.3) Suppose that $\mathcal{M}$ is locally presentable and $(\mathcal{L}, \mathcal{R})$ is an accessible wfs. Then there is a small category $I$ and a functor $D \colon I \to \mathcal{M}^\to$ such that $\mathcal{R}$ is precisely the class of maps admitting a lifting structure against $D$.


Note in particular that the class of maps admitting a lifting structure against $D$ is closed under retracts (see also Remark 3.4 in Rosický, and Lemma 2.30 in Riehl).


**Lemma 5** Let $\mathcal{M}$ be locally presentable and $I$ and $J$ small. Suppose we are given functors $D \colon I \to \mathcal{M}^\to$ and $E \colon J \to \mathcal{M}^\to$ together with a functor $F \colon I \to J$ making a commutative triangle. Then there is a morphism of algebraic weak factorisation systems from the awfs cofibrantly generated by $D$ to the awfs cofibrantly generated by $E$.


**Proof** As Riehl remarks in [*Algebraic model structures*](https://arxiv.org/pdf/0910.2733.pdf), this follows from freeness. There is another way to phrase this using reflections. Garner's argument, as presented in [*Understanding the small object argument*](https://link.springer.com/article/10.1007/s10485-008-9137-4) in fact produces reflections for a canonical functor $\mathbf{AWFS}(\mathcal{M}) \to \mathbf{CAT}/\mathcal{M}^\to$. In general, given a functor $G \colon \mathcal{C} \to \mathcal{D}$, a reflection of an object $D \in \mathcal{D}$ is an initial object of the comma category $(D \downarrow G)$. Given a morphism between two objects $D \to D'$ in $\mathcal{D}$, there is a unique map between the components of their reflections in $\mathcal{C}$ making a commutative square. Applying to this special case, any morphism in $\mathbf{CAT}/\mathcal{M}^\to$ gives rise to a morphism between reflections in $\mathbf{AWFS}(\mathcal{M})$.


**Theorem 6** Let $\mathcal{M}$ be locally presentable and $(\mathcal{C}, \mathcal{W}, \mathcal{F})$ an accessible model structure on $\mathcal{M}$. Then there is an algebraic model structure $\xi \colon (C, F^t) \to (C^t, F)$ on $\mathcal{M}$, where the class of maps admitting $F$-algebra structure is precisely $\mathcal{F}$ and the class of maps admitting $F^t$-algebra structure is precisely $\mathcal{F} \cap \mathcal{W}$.


**Proof** By Theorem 4, there are diagrams $D \colon I \to \mathcal{M}^\to$ and $E \colon J \to \mathcal{M}^\to$, with $I, J$ small categories such that $\mathcal{F}$ is the class of maps admitting a lifting structure against $D$ and $\mathcal{F}^t := \mathcal{F} \cap \mathcal{W}$ is the class of maps admitting a lifting structure against $E$. We take $(C^t, F)$ to be the awfs cofibrantly generated by $D$ and take $(C, F^t)$ to be the awfs cofibrantly generated by $D + E \colon I + J \to \mathcal{M}^\to$. By Lemma 5, the inclusion $D \to D + E$ induces a morphism of awfs's $\xi \colon (C^t, F) \to (C, F^t)$. By Theorems 2 and 4, $\mathcal{F}$ is precisely the class of maps admitting $F$-algebra structure. Now let $g$ be a map in $\mathcal{F}^t$. Then it also lies in $\mathcal{F}$, and so by Theorem 4 it admits lifting structures against both $D$ and $E$. Hence by Lemma 3 it admits a lifting structure against $D + E$. Hence it admits $F^t$-algebra structure by Theorem 2. The converse is straightforward using Theorem 2 again. Hence the wfs underlying $(C, F^t)$ is precisely $(\mathcal{C}, \mathcal{F}^t)$. We have now shown that the model structure underlying the algebraic model structure $\xi$ is precisely $(\mathcal{C}, \mathcal{W}, \mathcal{F})$.


