I was reading this paper by Bousfield on the localization of spectra. On page 5, Lemma 1.13, there's a rather small curious technical detail on wedge sum. We have for a limit ordinal $\lambda,B_{\lambda}=\bigcup_{\sigma<\lambda}B_{\sigma}$ where $0 =B_{0} \subset B_{1} \subset B_{2} \subset \cdots $. It says that there is a cofiber sequence:

$\bigvee_{s < \lambda} B_{s} \stackrel{1-i}\to \bigvee_{s<\lambda}B_{s} \to B_{\lambda}$

where $i$ is the wedge of the inclusions $B_{s} \hookrightarrow B_{s+1}$. How exactly is this a cofiber sequence. I get the rough idea that one is trying to collapse things in the wedge sum to get the union but I'm not sure how a formal proof would work. I hope someone can help me out with this.

  • 1
    $\begingroup$ It's in some sense dual to the first part of Lemma 1.8 in that paper, and it's also related to the fact that in an abelian category, you can write a colimit of objects $X_\alpha$ as the cokernel of a map between the coproduct of the $X_\alpha$s. (Lemma 1.8 is defining a homotopy limit, whereas this is supposed to be some sort of actual colimit. But in general in a triangulated category, this cofiber sequence is how you would define a (weak) colimit of a sequence of maps.) $\endgroup$ – John Palmieri May 7 '20 at 20:24
  • $\begingroup$ The philosophy is that if an $n$-cell in $B_\lambda$ appears in some $B_j$ and all $B_k$ with $k \geq j$, then the map $1-i$ will identify all of these $n$-cells in the wedge into a single one. $\endgroup$ – John Palmieri May 7 '20 at 20:27

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