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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.

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1 Answer

The condition you're looking for is called combinatoriality (and local presentability). A model category is combinatorial provided it satisfies some complicated conditions involving accessibility, but at least when the underlying category is a presheaf topos and the cofibrations are exactly the monomorphisms, we can actually systematically construct these model structures using the framework of Denis-Charles Cisinski in his book Les Préfaisceaux comme modèles des types d'homotopie (Astérisque 308). This, in particular gives us the combinatoriality of the model structure on simplicial sets essentially for free.

For a general reference on combinatoriality and Jeff Smith's theorem, I suggest you take a look at the papers Sheafifiable Homotopy Model Categories by Tibor Beke and On Left and Right Model Categories and Left and Right Bousfield Localizations by Clark Barwick. (The original theorem is due to Jeff Smith, but he has neglected to publish it (it was announced at a conference and the main ideas were given, I believe), although he has assured us (not me specifically!) many times that he is in the process of publishing a book).

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By the way, combinatoriality is the condition, and Jeff Smith's theorem is the theorem that proves al the nice things about it. –  Harry Gindi Feb 21 '11 at 15:20
Combinatoriality is for filtered stuff, not just ordinal stuff. Did you read the references I gave you? For instance, the statement about weak equivalences that you want is proven in the appendix to Jacob Lurie's Higher Topos Theory (I want to say in section 2.6). –  Harry Gindi Feb 22 '11 at 1:57
@Harry, the underlying category of symmetric spectra is not precisely "simplicial symmetric sequences" (rather it is modules over the sphere spectrum symmetric sequence). However, the category of symmetric spectra is locally presentable and the model structures constructed in, e.g. HSS, MMSS, Shipley's "A Convenient ..." paper are all cofibrantly generated, hence combinatorial. Although the combinatorial condition is the "right" condition for this sort of thing, for spectra, you don't need all this machinery. You might just want section 2 of Hovey-Palmieri-Strickland. –  Sam Isaacson Feb 23 '11 at 7:03
@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
@Harry, combinatorial usually means "locally presentable" and "cofibrantly generated"; did I not make that clear in my comment? Regarding model structures on symmetric spectra, you might add Schwede's excellent "Untitled book project ..." (found on his website) to a list of references. The abundance of model structures on symmetric spectra is a good thing. –  Sam Isaacson Feb 23 '11 at 15:35
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