Does any accessible model category come from an algebraic model category?

Question

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

Answer 1

For reference, here is a more detailed version of Riehl's argument.

Definition 1 (Garner, Understanding the small object argument, 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, 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, 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, this follows from freeness. There is another way to phrase this using reflections. Garner's argument, as presented in Understanding the small object argument 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 $(\mathcal{C}, \mathcal{W}, \mathcal{F})$.