Hi, I have a question on weak-equivalences of spectra.


More precisely, I wonder whether filtered colimits of weak-equivalences of spectra are again weak-equivalences of spectra. Here, spectra are in the sense of Bousfield-Friedlander, i.e. a sequence of pointed simplicial sets $(E\_0, E\_1, \cdots, )$ with the morphisms $\sigma\_n : S^1 \wedge E\_n \to E\_{n+1}$ for all $n \geq 0$. Of course, weak-equivalences of spectra are by definition the stable weak-equivalences.


After some efforts, I was able to find from a webpage that, if the spectra in question are all Kan spectra (spectra whose all levels are Kan simplicial sets), then this is indeed true. But, I have no idea how to proceed in general, partly because I do not think spectra are cofibrant in general.


I was informally told by someone that, in general, if we are working with a ``combinatorial'' model category, then such questions are likely be true. But, is it true that the category of spectra is combinatorial, and if that is the case, then does the main question hold?


I am fine with working with only Kan spectra if this question turns out to be negative, but since I didn't do my basic graduate school studies in algebraic topology, I hoped to know the answers for this question first for I thought the question might be easy for experts in the field. Thank you.



Are filtered colimits of weak-equivalences of spectra again weak-equivalences?
The answer is **yes**, but the reason is technical. 


The reason is that, if I understand well, you're asking whether weak equivalences are closed under *arbitrary* filtered colimits. These are $\aleph\_0$-filtered colimits, and there is a hierarchy of degrees of filtration parametrized by all infinite regular cardinals $\alpha$, being $\alpha=\aleph\_0$ the first one. 


In general combinatorial model categories, you know that there exists a big enough regular cardinal $\lambda$ such weak equivalences are closed under $\alpha$-filtered colimits for all $\alpha\geq \lambda$, but not necessarily for $\alpha<\lambda$. 


The category of spectra of simplicial sets is locally $\aleph\_0$-presentable and has sets of generating (trivial) cofibrations with $\aleph\_0$-presentable sources. Hence you can take $\lambda=\aleph\_0$.


See section 7 of


D. Dugger, Combinatorial model categories have presentations, Adv. Math. 164 (2001), 177–201,


and the appendix of


S. Schwede, Stable homotopy of algebraic theories, Topology 40 (2001), 1–41.


