When does a cofibrantly generated model category have this factorization property?

Question

Let $\mathcal{C}$ be a cofibrantly generated model category, which is generated by $I$ and $J$. According to the small object argument (Hovey Theorem 2.1.14) of cofibrantly generated model categories, each map $X\to Y$ in $\mathcal{C}$ could be factored as $$X\xrightarrow{i} Z \xrightarrow{r} Y,$$ where $i$ is a relative $I$-cell complex and $r$ is a weak equivalence. But now, I need the cofibrantly generated model category $\mathcal{C}$ to satisfy the extra following property:

For any finite $I$-cell complex $X$ in $\mathcal{C}$, the fold map $(\mathsf{id}_X,\mathsf{id}_X)\colon X\amalg X \to X$ could always be factored as $$X\amalg X \xrightarrow{(i_0,i_1)} X' \xrightarrow{r} X$$ where $(i_0,i_1)$ is a relative finite $I$-cell complex and $r$ is a weak equivalence.

(To avoid misunderstanding, a relative finite $ I $‑cell complex means that both the factorization and the coproduct of morphisms in $I$ are finite.)

I think this property is not established for every cofibrantly generated model categories, but I also think it is not a very strong restriction. So I was wondering if there are some "deeper meaning" of this property (in a more general view)? or some large classes of cofibrantly generated model categories (maybe with extra structure such as enrich structure) has this property?

Additionally, the category $\mathbf{Top}$ (or $\mathbf{SSet}$) is too special for an example of this property, since the "middle object" $X'=X\times [0,1]$ of such factorization is generated by the catesian product, and the finiteness of $X\times [0,1]$ also rely on the catesian product. And I also read Williamson's thesis Cylindrical model structures , it looks like it satisfies this property, but as I said I think this property may not depend on the "cylindrical structure".

Answer 1

I've encountered that condition a few time. Here is what I know about it:

If that property is satisfied for a model category $M$ then the full subcategory $C$ of finite $I$-cell complex is itself a Brown category of cofibrant object.

So they both have $\infty$-category attached to them: $h_\infty(M)$ which is locally presentable and $h_\infty(C)$ which his finitely co-complete. If we add the assumption that all maps in $I$ are between finitely presentable objects of $C$, then $h_\infty(M)$ is $\omega$-presentable with

$$h_\infty(M) \simeq Ind(h_\infty(C)) $$

(the ind completion)

Conversely, given any small Brown category of cofibrant objects $C$ you can build canonically a $\omega$-combinatorial left semi-model category $M$ such that $C$ identitfies with the finite $I$-cell complex in $M$. (and hence $M$ is a model for the infinity category cal ind completion of $C$).

Another remark is that all this can be generalized from "finite" to "$\lambda$-small" for any $\lambda$ and if $M$ is a combinatorial model category then there is $\lambda$ such that it satisfies these conditions (i.e. $\lambda$-small $I$-complex have cylinder that are relative $\lambda$-small complex ).

Some of these claims appears in this paper of mine, the rest can be justified from there (except the clain about $\infty$-categories, that will be in an upcoming work).