# Do homotopy groups “always” commute with filtered colimits?

It is well-known that homotopy groups, of, say, simplicial sets, commute with filtered colimits.

However, I could not find a reference for an analogous result for homotopy groups of spectra, or, under which hypothesis the same result would hold for an "arbitrary" simplicial model category.

More precisely, let $\cal{M}$ be a simplicial model category. For a fibrant object $X \in \cal{M}$ its homotopy groups with coefficients in a cofibrant object $W\in \cal{M}$ may be defined as

$$\pi_n (X; W) = [\Sigma^nW, X] = \pi_n \mathrm{map}(W,X) \ ,$$

where $\mathrm{map}$ denotes the simplicial mapping space from $W$ to $X$.

So my first question is: which hypothesis do I have to assume for $W$ to obtain an isomorphism

$$>\mathrm{colim}_i \pi_n (X_i;W) = \pi_n (\mathrm{colim}_i X_i;W) \ ? >$$

And the second one: in which kind of model category such an isomorphism holds for every cofibrant object $W$ -or, at least, for "sufficiently" many cofibrant objects $W$?

The reason behind my question is the following (and explains the meaning of that "sufficiently"): I have a filtered category $I$, functors $X_\bullet, Y_\bullet : I \longrightarrow {\cal M}\_f$ and a natural transformation $f_\bullet : X_\bullet \longrightarrow Y_\bullet$, such that, for every cofibrant object $W$, $f_\bullet$ induces isomorphisms

$$\mathrm{colim}_i \pi_n (X_i ; W) = \mathrm{colim}_i \pi_n (Y_i ; W) \ ,$$

for every $n$. And I want to conclude that the induced map between the colimits

$$\mathrm{colim}_i X_i \longrightarrow \mathrm{colim}_i Y_i$$

is a weak equivalence. Which would be true if

1. I could commute colimits and homotopy groups, at least for

2. "enough" cofibrant objects $W$ -in case of simplicial sets, one point $W = *$ is enough.

I suspect the answer involves words like "smallness / compactness" and "cellular model category". For instance an answer like: "You can do that in no matter what simplicial cellular model category" -in which every cofibrant object is compact- would be fine. Nevertheless, as long as I can understand, commutations like

$$\mathrm{colim}_i {\cal M} (W, X_i ) = {\cal M} (W, \mathrm{colim}_i X_i)$$

hold for $W$ small and $\lambda$-sequences; that is, when the domain of the functor $X : \lambda \longrightarrow \cal{M}$ is an ordinal; in particular, a totally ordered set, which my filtered $I$ needs not to be.

So any references of a result along these lines, even just for spectra, are welcome.

• @Sam: Yes, I realize that it's not the same underlying category, but the idea is similar to how the category of simplicial $A$-modules is combinatorial for any simplicial Commutative Ring $A$. By the way, cofibrant generation absolutely does not imply combinatoriality. Combinatoriality is strictly stronger (unless you're suggesting that we take Vopenka's principle to be true). – Harry Gindi Feb 23 '11 at 7:38