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This question arises when I am reading the last two Chapter of Hirschhorn's "Model categories and their localizations"

In Part (2) of Theorem 19.8.4 of that book it says

If $(\bf{\Delta},\mathcal{M})$ is a Reedy framed diagram category structure on the category of cosimplicial objects in $\mathcal{M}$ and $\bf X$ is a cosimplicial object in $\mathcal{M}$, then the total object $\text{Tot}\bf X$ of $\bf X$ is naturally weakly equivalent to the homotopy limit $\text{holim} \bf X$ of $\bf X$.

I notice that in Part (1) of the same theorem there is a dual statement on homotopy colimit of simplicial object but it require that $\bf X$ is Reedy cofibrant.

My question is: could we really get rid of the fibrancy requirement if we want to show that the totalization of a cosimplicial object is weakly equivalent to the homotopy limit?

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  • $\begingroup$ I expect that it is a typographical error. $\endgroup$
    – Zhen Lin
    Commented Jul 3, 2015 at 14:18
  • $\begingroup$ @ZhenLin I also expected that. But the same thing happens in Theorem 19.8.2 (but not in Theorem 19.8.3). $\endgroup$
    – Zhaoting Wei
    Commented Jul 3, 2015 at 16:14
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    $\begingroup$ A list of errata is maintained by Phil here, but nothing yet in chapter 19: www-math.mit.edu/~psh/MCATL-errata-2015-01-11.pdf $\endgroup$
    – David White
    Commented Jul 3, 2015 at 16:33
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    $\begingroup$ It often depends on what model you take for the holim. Nowadays it seems customary for people to implicitly fibrantly replace. If you want to know about the state of the art on these things I recommend Riehl's book, chapters 5 and 6. On page 71 she describes a setting where fibrant replacement is necessary, but I am not making this an answer because I don't have an explicit counterexample at hand (and not the time to think about it right now). I agree with Zhen Lin that I do not expect a positive answer to your question. $\endgroup$
    – David White
    Commented Jul 3, 2015 at 16:42

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