I have the following situation. $M$ is a combinatorial model category, or if you like a locally presentable $(\infty,1)$-category. I have a set of maps $S$ and I let $C$ be the class of maps generated from maps in $S$ via directed colimits. This means every $f\in C$ can be written as a directed colimit of $f\_i \in S$, and because $M$ is (co)complete this means the domain of $f$ and is the colimit of the domains of the $f\_i$ and similarly for the codomain of $f$.



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> Is it true that $f$ is in $S$-cell (i.e. it's a retract of some transfinite compositions of pushouts of maps in $S$)? More generally, do we have $S$-cell $=$ $C$-cell?
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A word of caution. Various facts about model categories will tell you that $S$-cell is closed under directed colimits and maybe even filtered colimits. But it seems to me that's for colimits taken in $M$ not in $Arr(M)$, i.e. it's saying that if we have a chain $X\_0\to X\_1\to \dots$ and each map $X\_i \to X\_{i+1}$ is in $S$-cell then the colimit $X\_0\to X\_\lambda$ is in $S$-cell. This is not what I'm asking for. I need a ladder $f\_0\to f\_1\to \dots$ and to know the colimit $f$ is in the same class $S$-cell which all the elements of the directed system are in. I'd like this for any directed colimit of the $f\_i$ but I'm willing to accept it for a colimit of the sort just drawn, i.e. a sequential colimit. At least then I'd have an intuition for whether or not this is true.



Directed colimits of maps in a combinatorial model category
The answer is 'no' (sorry I didn't realize this before!)


I remembered someone showing me a weird counterexample in model categories a couple weeks ago (elaborated on from this paper by Rosicky: <http://www.math.muni.cz/~rosicky/papers/comb2.pdf>)


and it seems to do the trick here:


Consider the category ${\bf Pos}$ of posets. This is certainly presentable. Let $C$ denote the collection of split monomorphisms (which is weakly saturated and generated by split monomorphisms between finite posets). Then we can define a combinatorial model structure on $\{\bf Pos}$ by taking *all* morphisms as weak equivalences (this satisfies the conditions of, say, Higher Topos Theory A.2.6.7. or one can see this directly). However, $C$ is not closed under sequential colimits because, for example, the non-split monomorphism arrow $\omega \rightarrow \omega \cup \infty$ is the sequential limit of the arrows $[n] \hookrightarrow [n] \cup \infty$, each of which is split.


P.S. What is your distinction between directed colimits and filtered colimits (this terminology has always confused me...)? The way you're using the word makes it seem like "directed" is the same as "finitely filtered". 


