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Let $\lambda$ be a limit ordinal, and let $F: \lambda\to \mathcal{T}_*$ be a diagram of pointed spaces with shape $\lambda$. Write $X = F(0)$ and $Y = \mathrm{hocolim} F$. I believe it to be true (I have a sketch of a proof) that if

  • every $F(\alpha)\to F(\beta)$ is a the inclusion of a sub-CW-complex, and
  • for every $\beta < \lambda$, $F(\beta+1)\hookrightarrow Y$ is homotopic, by a homotopy constant on $F(\beta)$, to a map that factors through $F(\beta)$

then the induced map $X\to Y$ is a homotopy equivalence.

This is certainly very classical in the case $\lambda = \omega$, but I would like a reference for the general result.

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    $\begingroup$ In your first condition, are you fixing a single CW structure on each $F(\alpha)$, or do you allow yourself do choose different CW structures depending on which map in or out of $F(\alpha)$ you're considering? $\endgroup$
    – Dan Ramras
    Commented Jul 1, 2019 at 4:29
  • $\begingroup$ Always the same structure $\endgroup$
    – Jeff Strom
    Commented Jul 1, 2019 at 11:53

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